{"id":264,"date":"2023-07-10T06:09:02","date_gmt":"2023-07-10T06:09:02","guid":{"rendered":"https:\/\/mathority.org\/de\/liniengleichungen-alle-formeln-beispiele-geloste-ubungen\/"},"modified":"2023-07-10T06:09:02","modified_gmt":"2023-07-10T06:09:02","slug":"liniengleichungen-alle-formeln-beispiele-geloste-ubungen","status":"publish","type":"post","link":"https:\/\/mathority.org\/de\/liniengleichungen-alle-formeln-beispiele-geloste-ubungen\/","title":{"rendered":"Liniengleichungen"},"content":{"rendered":"<p>Hier finden Sie die Formeln f\u00fcr alle Arten von Geradengleichungen. Dar\u00fcber hinaus k\u00f6nnen Sie Beispiele f\u00fcr deren Berechnung sehen und zus\u00e4tzlich mit gel\u00f6sten \u00dcbungen zu den Geradengleichungen \u00fcben. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuales-son-todas-las-ecuaciones-de-la-recta\"><\/span> Wie lauten alle Gleichungen der Geraden?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Denken Sie daran, dass die mathematische Definition einer Linie eine Menge aufeinanderfolgender Punkte ist, die in derselben Richtung ohne Kurven oder Winkel dargestellt werden.<\/p>\n<p> Um also jede Gerade in der Ebene (im R2) analytisch auszudr\u00fccken, verwenden wir die Gleichungen der Geraden, und um sie zu finden, braucht man nur einen Punkt, der zur Geraden geh\u00f6rt, und den Richtungsvektor dieser Geraden. Mit nur diesen beiden geometrischen Elementen k\u00f6nnen Sie absolut alle verschiedenen Gleichungen der Geraden finden, die wie folgt lauten:<\/p>\n<p> <strong>Die Gleichungen der Geraden sind die Vektorgleichung, die parametrischen Gleichungen, die kontinuierliche Gleichung, die implizite (oder allgemeine) Gleichung, die explizite Gleichung, die Punkt-Steigungs-Gleichung und die kanonische (oder segmentale) Gleichung.<\/strong><\/p>\n<p> Alle Arten von Liniengleichungen haben das gleiche Ziel: eine Linie mathematisch darzustellen. Aber jede Geradengleichung hat ihre eigenen Eigenschaften und daher ist es je nach Problem besser, die eine oder die andere zu verwenden. <\/p>\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equations-de-la-droite-1.webp\" alt=\"Gleichungen einer Geraden pdf\" width=\"287\" height=\"273\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Nachdem wir das Konzept der Liniengleichungen kennengelernt haben, gehen wir nun dazu \u00fcber, die Eigenschaften jedes einzelnen Typs von Liniengleichungen im Besonderen zu analysieren. Unten finden Sie eine detaillierte Erkl\u00e4rung der verschiedenen Arten von Gleichungen in der Zeile. Wenn Sie m\u00f6chten, k\u00f6nnen Sie jedoch direkt zum Ende der <a href=\"https:\/\/mathority.org\/de\/liniengleichungen,-alle-formeln,-beispiele,-geloste-ubungen\/\">\u00dcbersichtstabelle mit den Formeln aller Gleichungen in der Zeile<\/a> gehen. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-vectorial-de-la-recta\"><\/span> Vektorgleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Ja<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist der Richtungsvektor der Geraden und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> ein Punkt, der nach rechts geh\u00f6rt:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a5a9724c5deabef496a75b00995419d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= (\\text{v}_1,\\text{v}_2) \\qquad P(P}_1,P_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Die <strong>Formel f\u00fcr die Vektorgleichung der Geraden<\/strong> lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6e64023d7dbfb100dc641c09e202e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      (x,y)=(P_1,P_2)+t\\cdot (\\text{v}_1,\\text{v}_2) \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> sind die kartesischen Koordinaten eines beliebigen Punktes auf der Linie.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d38a31ec1eb0a45c9ee8e1b143e3b4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c78cc5579163a0956b9462599d75b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Koordinaten eines bekannten Punktes, der Teil der Linie ist<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6773414e1c04325d3dcb0a9f1e232f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(P}_1,P_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16a61eafb9e0a7b88b98a7fffd74c09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43a68c72834dd1643b28f72554b27956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Komponenten des Richtungsvektors der Geraden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52295cf8445bb05e7ea88d57dca521e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(\\text{v}_1,\\text{v}_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist ein Skalar (eine reelle Zahl), dessen Wert von jedem Punkt auf der Linie abh\u00e4ngt.<\/li>\n<\/ul>\n<p> Es ist die Vektorgleichung der Geraden in der Ebene, also bei der Arbeit mit Punkten und Vektoren von 2 Koordinaten (im R2). Wenn wir jedoch Berechnungen im Raum (im R3) durchf\u00fchren w\u00fcrden, m\u00fcssten wir der Geradengleichung eine zus\u00e4tzliche Komponente hinzuf\u00fcgen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ef53596406b2fe36258a0421c91336b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_1,P_2,P_3)+t\\cdot (\\text{v}_1,\\text{v}_2,\\text{v}_3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"288\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuaciones-parametricas-de-la-recta\"><\/span> Parametrische Gleichungen der Linie<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Die parametrischen Gleichungen einer Geraden lassen sich aus ihrer Vektorgleichung ermitteln:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16e43c9d65f7fae5b0e20a3caca1df38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1,P_2)+t\\cdot (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Wir multiplizieren zun\u00e4chst den Parameter<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> durch den Richtungsvektor von rechts:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fdb6864861b77af0532f9a000fe566d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1,P_2)+ (t\\cdot\\text{v}_1,t\\cdot\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Als n\u00e4chstes f\u00fcgen wir die X- und Y-Koordinaten hinzu:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbd7cd2cffb8ff3378d0a03949644e0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1+t\\cdot\\text{v}_1,P_2+t\\cdot\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Und schlie\u00dflich, indem wir jede Variable einzeln l\u00f6schen, erhalten wir die parametrischen Gleichungen der Geraden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46f6cdd4b1d1a92d038d140904abd119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle \\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\end{cases} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> sind die kartesischen Koordinaten eines beliebigen Punktes auf der Linie.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d38a31ec1eb0a45c9ee8e1b143e3b4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c78cc5579163a0956b9462599d75b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Koordinaten eines bekannten Punktes, der Teil der Linie ist<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6773414e1c04325d3dcb0a9f1e232f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(P}_1,P_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16a61eafb9e0a7b88b98a7fffd74c09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43a68c72834dd1643b28f72554b27956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Komponenten des Richtungsvektors der Geraden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52295cf8445bb05e7ea88d57dca521e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(\\text{v}_1,\\text{v}_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist ein Skalar (eine reelle Zahl), dessen Wert von jedem Punkt auf der Linie abh\u00e4ngt.<\/li>\n<\/ul>\n<p> Nach wie vor sind dies die parametrischen Gleichungen der Linie in der Ebene (in R2), aber um die parametrischen Gleichungen der Linie im Raum (in R3) zu finden, m\u00fcsste eine weitere Gleichung f\u00fcr die dritte Variable Z hinzugef\u00fcgt werden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e31f05449ce57a8af9ae4dda38535013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\\\[1.7ex] z=P_3+t\\cdot\\text{v}_3\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-continua-de-la-recta\"><\/span>Kontinuierliche Gleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Die kontinuierliche Gleichung jeder Geraden kann aus ihren parametrischen Gleichungen abgeleitet werden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-708dbb33878e2bab0dcc94c84f6ab670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Wenn wir die Einstellung l\u00f6schen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> Aus jeder Parametergleichung erhalten wir die folgenden Ausdr\u00fccke:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50f7c5405a4fc4f6faa3b8f4b651fb97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t =\\cfrac{x-P_1}{\\text{v}_1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"83\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de8a9e455480e01bf5166f9519430491_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t =\\cfrac{y-P_2}{\\text{v}_2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"83\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> E Indem wir die beiden resultierenden Gleichungen gleichsetzen, erhalten wir die kontinuierliche Gleichung der Geraden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26c55cd229e56a297715f1c05891a523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t= t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-913e6797e350e331ce17df6b5c074f91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> Kurz gesagt lautet die <strong>kontinuierliche Gleichung der Geraden<\/strong> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7063ed965532bc4df04315115aa10bdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> sind die kartesischen Koordinaten eines beliebigen Punktes auf der Linie.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d38a31ec1eb0a45c9ee8e1b143e3b4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c78cc5579163a0956b9462599d75b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Koordinaten eines bekannten Punktes, der Teil der Linie ist<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6773414e1c04325d3dcb0a9f1e232f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(P}_1,P_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16a61eafb9e0a7b88b98a7fffd74c09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43a68c72834dd1643b28f72554b27956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> sind die Komponenten des Richtungsvektors der Geraden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52295cf8445bb05e7ea88d57dca521e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(\\text{v}_1,\\text{v}_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p> Diese Formel gilt f\u00fcr die kontinuierliche Gleichung der Linie beim Arbeiten in zwei Dimensionen (in 2D). Wenn wir jedoch Operationen in drei Dimensionen (3D) durchf\u00fchren w\u00fcrden, m\u00fcssten wir der Liniengleichung eine zus\u00e4tzliche Komponente hinzuf\u00fcgen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a090d35f6f6edef6dfff9c124862a49a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2}= \\cfrac{z-P_3}{\\text{v}_3}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"202\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-implicita-o-general-de-la-recta\"><\/span> Implizite oder allgemeine Gleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Ja<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist der Richtungsvektor der Geraden und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> ein Punkt, der nach rechts geh\u00f6rt:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a5a9724c5deabef496a75b00995419d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= (\\text{v}_1,\\text{v}_2) \\qquad P(P}_1,P_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Die Formel f\u00fcr die <strong>implizite, allgemeine oder kartesische Gleichung der Geraden<\/strong> lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acd74645ce35f9b771269d09bb1e0b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      Ax+By+C=0 \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> sind die kartesischen Koordinaten eines beliebigen Punktes auf der Linie.<\/li>\n<li> der Koeffizient\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist die zweite Komponente des Richtungsvektors der Geraden:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8aae57bb8c0ba7650d53c865bdf4855a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\text{v}_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -3px;\"><\/p>\n<\/li>\n<li> der Koeffizient\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist die erste Komponente des Richtungsvektors mit ge\u00e4ndertem Vorzeichen:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a42f7e7fc1557de4f36ee335a3ff6c64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B=-\\text{v}_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: -3px;\"><\/p>\n<\/li>\n<li> der Koeffizient\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> wird durch Ersetzen des bekannten Punktes berechnet<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> in der Geradengleichung.<\/li>\n<\/ul>\n<p> Die Formel, die implizite Gleichung einer Geraden, kann auch durch Multiplikation der Br\u00fcche der stetigen Gleichung erhalten werden. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-explicita-de-la-recta\"><\/span> Explizite Gleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Die Formel f\u00fcr die <strong>explizite Gleichung der Geraden<\/strong> lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-70bd24576c0a37b64c5731799e67083e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      y=mx+n \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist die Steigung der Geraden.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> sein y-Achsenabschnitt, also die H\u00f6he, in der er die Y-Achse schneidet.<\/li>\n<\/ul>\n<p> Im folgenden Abschnitt erfahren Sie, wie die Parameter ermittelt werden<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> der Geraden Eine andere M\u00f6glichkeit, die explizite Gleichung zu finden, besteht jedoch insbesondere darin, die implizite Gleichung zu verwenden. Daf\u00fcr muss das Unbekannte gel\u00f6st werden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> der impliziten Gleichung.<\/p>\n<h4 class=\"wp-block-heading\"> Bedeutung der Parameter m und n<\/h4>\n<p> Wie wir in der Definition der expliziten Gleichung der Geraden gesehen haben, ist der Parameter<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist die Steigung der Geraden und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> sein y-Achsenabschnitt. Aber was bedeutet das? Sehen wir uns dies anhand der grafischen Darstellung einer Linie an: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-explicite-d-une-ligne.webp\" alt=\"Wie lautet die explizite Gleichung der Geraden y=mx+b?\" class=\"wp-image-1455\" width=\"339\" height=\"339\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Der Begriff unabh\u00e4ngig<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-565fee0d356edf7fb1f49b6e7eec8e61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{n}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> <strong>ist der Schnittpunkt der Linie mit der Computerachse<\/strong> (OY-Achse). Zum Beispiel in der Grafik oben<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist gleich 1, da die Linie die y-Achse bei y=1 schneidet.<\/p>\n<p> Andererseits der Begriff<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f26b1f086c6ad942d7c0dac86a8338fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{m}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> <strong>gibt die Steigung der Geraden an<\/strong> , also ihre Neigung. Wie Sie in der Grafik sehen k\u00f6nnen,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist gleich 2, da die Linie um 2 vertikale Einheiten f\u00fcr 1 horizontale Einheit ansteigt.<\/p>\n<p> Offensichtlich nimmt die Funktion zu, wenn die Steigung positiv ist (steigt), wenn die Steigung negativ ist, nimmt die Funktion ab (sinkt).<\/p>\n<h5 class=\"wp-block-heading\"> Berechnen Sie die Steigung einer Geraden<\/h5>\n<p> Sobald wir die Steigung einer Geraden genau kennen, schauen wir uns an, wie sie berechnet wird. Es gibt also drei verschiedene M\u00f6glichkeiten, die Steigung einer Geraden numerisch zu bestimmen:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Gegeben seien zwei unterschiedliche Punkte auf der Geraden\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-99906702500e51b12e2859cc804a7b57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(x_1,y_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-460a66d684215738da922dc45a35aed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_2(x_2,y_2),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<p> Die Steigung der Geraden ist gleich:<\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ca826248e812d4f19056960777cb00f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{\\Delta y}{\\Delta x} = \\cfrac{y_2-y_1}{x_2-x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"150\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Ja\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-867fb10d1409b3d95ff447f6a095219d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> ist der Richtungsvektor der Geraden, ihre Steigung ist:<\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60d899a76c2b7588e60dc3734a47019f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{\\text{v}_2}{\\text{v}_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"59\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Ja\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> ist der Winkel, den die Linie mit der Abszissenachse (X-Achse) bildet, die Steigung der Linie entspricht dem Tangens dieses Winkels: <\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c76cc82b1d172b2b5af3b053752befac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}(\\alpha )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/ol>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-l-equation-explicite-d-une-ligne.webp\" alt=\"Formel f\u00fcr die explizite Gleichung der Geraden\" class=\"wp-image-1465\" width=\"288\" height=\"356\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-punto-pendiente-de-la-recta\"><\/span> Punkt-Steigungsgleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Die Formel f\u00fcr die <strong>Punkt-Steigungsgleichung der Geraden<\/strong> lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d1f485a8e43f9f81d8711d2f17dac20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      y-P_2=m(x-P_1) \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Gold:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> ist die Steigung der Geraden.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4c0be0b31844a0cd94ce4d5ea2a7256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1, P_2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\"><\/p>\n<p> sind die Koordinaten eines Punktes auf der Geraden <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a813701c043bb25e074ddaba52d46a0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(P_1,P_2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-canonica-o-segmentaria-de-la-recta\"><\/span> Kanonische oder segmentale Gleichung der Geraden<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Obwohl diese Variante der Geradengleichung weniger bekannt ist, kann die kanonische Geradengleichung aus den Schnittpunkten der Geraden mit den kartesischen Achsen erhalten werden.<\/p>\n<p> Die beiden Schnittpunkte mit den Achsen einer gegebenen Geraden seien:<\/p>\n<p class=\"has-text-align-center\"> Mit der X-Achse schneiden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73f7f9618f43f69c0d8a68ff9b47ffef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> Schnitt mit Y-Achse:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aee2e1bda5b37d0b02db636b7d6a73e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Die <strong>Formel f\u00fcr die kanonische Geradengleichung<\/strong> lautet: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f6882981d96c9f3eb383d6a005eca81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\cfrac{x}{a}+\\cfrac{y}{b} = 1  \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-canonique-segmentaire-ou-symetrique-d-une-ligne.webp\" alt=\"Linienrechner-Gleichungen\" class=\"wp-image-3261\" width=\"297\" height=\"298\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> In der Mathematik wird die kanonische Geradengleichung auch Segmentgleichung oder symmetrische Gleichung genannt.<\/p>\n<p> Andererseits die Koeffizienten<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> Sie k\u00f6nnen auch aus der allgemeinen Geradengleichung mithilfe der folgenden Formeln ermittelt werden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02f4d03229cc8bd79a81b676a8132f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+C=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ebb3b443a8362ad9f023f8a2df2f17b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a = -\\cfrac{C}{A} \\qquad \\qquad b = -\\cfrac{C}{B}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"196\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\" id=\"tabla-resumen-formulas-de-todas-las-ecuaciones-de-la-recta\"><span class=\"ez-toc-section\" id=\"todas-las-ecuaciones-de-la-recta-formulas\"><\/span> Alle Gleichungen der Geraden (Formeln)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Zusammenfassend finden Sie hier eine Tabelle, die die Formeln aller Gleichungen der Geraden zeigt: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/toutes-les-equations-des-formules-de-ligne.webp\" alt=\"\" class=\"wp-image-3276\" width=\"581\" height=\"451\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-las-ecuaciones-de-la-recta\"><\/span> Beispiel f\u00fcr die Berechnung von Geradengleichungen<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nachdem wir nun die gesamte Erkl\u00e4rung der Geradengleichung gesehen haben, sehen wir uns an, wie ein typisches Problem von Geradengleichungen gel\u00f6st wird:<\/p>\n<ul>\n<li> Finden Sie alle Gleichungen der durch den Punkt bestimmten Geraden\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> und der Vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7fe319cb0fecedf2052e6c1e4c856733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ffa6488af1fcaf91bd9e53fd9133451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,-1) \\qquad \\qquad \\vv{\\text{v}}=(2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Zun\u00e4chst finden wir die Vektorgleichung der Geraden aus ihrer Formel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16e43c9d65f7fae5b0e20a3caca1df38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1,P_2)+t\\cdot (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Setzen Sie einfach die Koordinaten des Punktes und des Vektors in die Formel ein:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-499cd8e60d74a468d0f312e9cd346a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{(x,y)=(3,-1)+t\\cdot (2,4)} \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"534\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Zweitens finden wir die parametrischen Gleichungen der Geraden durch die entsprechende Formel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2e6878c4d9b80337639f5fa7728a9f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4690a2ab033a4016f2d16b9554ddea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\begin{cases} \\bm{x=3+2t} \\\\[1.7ex] \\bm{y=-1+4t} \\end{cases} \\quad \\vphantom{\\cfrac{\\cfrac{1}{2}}{\\cfrac{1}{2}}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Und wir bestimmen auch die stetige Geradengleichung mit ihrer Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-913e6797e350e331ce17df6b5c074f91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2f5db81c1d59dde56d49b2fbb142f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-3}{2}=\\cfrac{y-(-1)}{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"134\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f16821949f92a6284906d5a334bcc09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\cfrac{\\bm{x-3}}{\\bm{2}}\\bm{=}\\cfrac{\\bm{y+1}}{\\bm{4}}\\quad \\vphantom{\\Biggl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"434\" style=\"vertical-align: -28px;\"><\/p>\n<\/p>\n<p> Wie Sie gesehen haben, sind Vektorgleichungen, parametrische und kontinuierliche Gleichungen einfach zu berechnen, Sie m\u00fcssen lediglich die entsprechenden Formeln verwenden.<\/p>\n<p> Kommen wir nun dazu, die allgemeine (oder implizite) Gleichung der Geraden zu finden. Dazu kreuzen wir die beiden Br\u00fcche der stetigen Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a751eccdd3f40ccfa5794b381a5e89f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot (x-3)= 2 \\cdot (y+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b811aa5db0504964c34ad20afa3d236b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x-12= 2y+2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"130\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0a9ad69e0178c30b6e64e3c831d9c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x-12-2y-2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"161\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a3e29454fce63f0dfdfce4af94f1a12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{4x-2y-14=0}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"466\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Jetzt k\u00f6nnen wir die explizite Gleichung der Geradenl\u00f6sung f\u00fcr die Unbekannte bestimmen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> der impliziten Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7353c2555925ed510b4981154f047e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x-2y-14=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-369e9cd3411a3f7feb8092114cd7ac46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2y=-4x+14\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-efc35d0e90899ca1c77e78a312bbf9f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-4x+14}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7dc1790cfc0976eb2b7affd9b541ea56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{y=2x-7}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"418\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Daher ist die Steigung der Geraden gleich 2 (Term, der die unabh\u00e4ngige Variable begleitet).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> ).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1ee3bb14bbe97a1114d697f8b45a9f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Und damit k\u00f6nnen wir die Punkt-Steigungsgleichung der Geraden mit ihrer Formel berechnen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f0c8bfe8364c4962a61ff66ab943aeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-P_2=m(x-P_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e3eec13b49d25de2c2f630cad81f4ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-(-1)=2(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3fea9b91fd8324deef9bae6501b30b6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{y+1=2(x-3)}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"462\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Um schlie\u00dflich die Segmentgleichung der Geraden zu finden, berechnen wir ihre Schnittpunkte mit den Achsen OX und OY und wenden dann ihre Formel an:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9729cef93354933e4dcf55d23f640e45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2x-7\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-53\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Schnittpunkt mit der Abszissenachse (X-Achse)<\/span> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b58ae9c2cbc8637b603a8deb159c2ccb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0=2x-7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-245f8be211088d9023ae23ea593765a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2x=-7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"78\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68d1225afc848d605b122d88f9b9759d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-7}{-2} = \\cfrac{7}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"101\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd7ca037d0505b02f143c65891bfd911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\frac{7}{2}, 0\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"50\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Schnittpunkt mit der y-Achse (Y-Achse)<\/span> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3847d07e55ef57d0a7a39cc7b79f1c03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2\\cdot 0-7\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c15f100d859ec9077a43994ca473b018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-7\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5c4b18318739a0269d0ba45618ee45f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(0,-7\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2f3b0119758235dab9c6000508936ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{a}+\\cfrac{y}{b} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0acb3effdbe516cb8f1dc3ede8eca716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\cfrac{\\bm{x}}{\\frac{\\bm{7}}{\\bm{2}}}+\\cfrac{\\bm{y}}{\\bm{-7}} \\bm{= 1} \\quad \\vphantom{\\Biggl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"422\" style=\"vertical-align: -28px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-de-la-recta-que-pasa-por-dos-puntos\"><\/span> Geradengleichung, die durch zwei Punkte verl\u00e4uft<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ein weiteres sehr h\u00e4ufiges Problem bei Liniengleichungen besteht darin, die Gleichung der Linie zu finden, die durch zwei gegebene Punkte bestimmt wird. Obwohl wir mit den 2 Punkten und dann der Gleichung den Richtungsvektor der Linie berechnen k\u00f6nnen, stellen wir Ihnen im Folgenden eine Formel zur Verf\u00fcgung, mit der Sie direkt und einfach die Gleichung dieser Linie finden k\u00f6nnen.<\/p>\n<p> Betrachten Sie zwei Punkte, die auf einer Linie liegen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37c795a7dbd872ca4e96199d5335efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(x_1,y_1) \\qquad \\qquad P_2(x_2,y_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"220\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Die <strong>Formel zur Ermittlung der Geradengleichung aus ihren beiden Punkten<\/strong> lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a249bd55d016ac1e7f34f42de22d6e99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      y-y_1= \\cfrac{y_2-y_1}{x_2-x_1} (x-x_1) \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Mit dieser Formel k\u00f6nnen wir die Punkt-Steigungsgleichung der Geraden direkt berechnen, wenn wir zwei Punkte erhalten, durch die die Gerade verl\u00e4uft. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-las-ecuaciones-de-la-recta\"><\/span> Probleme mit Geradengleichungen gel\u00f6st<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u00dcbung 1<\/h3>\n<p> Finden Sie die Vektorgleichung, die parametrischen Gleichungen und die kontinuierliche Gleichung der durch den Punkt definierten Linie<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> und sein Richtungsvektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7fe319cb0fecedf2052e6c1e4c856733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Sei beides: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dcc97a260264762a15a9baa7cf40f61b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(0,3) \\qquad \\qquad \\vv{\\text{v}}=(-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zuerst berechnen wir die Vektorgleichung der Geraden aus ihrer Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16e43c9d65f7fae5b0e20a3caca1df38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1,P_2)+t\\cdot (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76d743d0188e28e2dbb0ad828a671b2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{(x,y)=(0,3)+t\\cdot (-1,5)} \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"534\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Anschlie\u00dfend ermitteln wir die parametrischen Gleichungen der Geraden mithilfe der entsprechenden Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2e6878c4d9b80337639f5fa7728a9f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a734c32ae40ca816c19b895e54916eb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x=0+t\\cdot (-1) \\\\[1.7ex] y=3+t\\cdot 5\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"132\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bff16cf5ab85c87d8a866a2d74ea2a31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\begin{cases} \\bm{x=-t} \\\\[1.7ex] \\bm{y=3+5t} \\end{cases} \\quad \\vphantom{\\cfrac{\\cfrac{1}{2}}{\\cfrac{1}{2}}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich bestimmen wir die stetige Geradengleichung mit der zugeh\u00f6rigen Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-913e6797e350e331ce17df6b5c074f91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30b363ea4f3d08cad83314f97a489b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-0}{-1}=\\cfrac{y-3}{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"106\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7bf9c947876ffb8003c437997c799f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\cfrac{\\bm{x}}{\\bm{-1}}\\bm{=}\\cfrac{\\bm{y-3}}{\\bm{5}}\\quad \\vphantom{\\Biggl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"415\" style=\"vertical-align: -28px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 2<\/h3>\n<p> Finden Sie die implizite Gleichung, die explizite Gleichung und die Punkt-Steigungsgleichung der durch den Punkt bestimmten Linie<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> und sein Richtungsvektor ist <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7fe319cb0fecedf2052e6c1e4c856733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a31faba9bf39a58f03087eaea99c0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-4,3) \\qquad \\qquad \\vv{\\text{v}}=(2,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Die Formel f\u00fcr die implizite Gleichung der Geraden lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02f4d03229cc8bd79a81b676a8132f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+C=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wir m\u00fcssen also die Koeffizienten A, B und C finden. Die Unbekannten A und B erh\u00e4lt man aus den Koordinaten des Richtungsvektors der Geraden, weil die folgende Gleichheit immer verifiziert ist:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-caffe051bad6b2835981c69786d9c98f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= (-B,A)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Folglich ist der Koeffizient A die zweite Koordinate des Vektors und der Koeffizient B ist die erste Koordinate des Vektors mit ge\u00e4ndertem Vorzeichen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9357fbcba6acde824f0fa1cc3e389a0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{c}\\vv{\\text{v}}= (-B,A) \\\\[2ex] \\vv{\\text{v}}= (2,6) \\end{array} \\right\\}\\longrightarrow \\begin{array}{l}A=6 \\\\[2ex] B=-2 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"226\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daher m\u00fcssen wir nur den Koeffizienten C ermitteln. Dazu m\u00fcssen wir den Punkt, von dem wir wissen, dass er zur Geraden geh\u00f6rt, in deren Gleichung einsetzen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2db3b6a5db31cf3a61fcd309886826b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-4,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e82cb96c9d0a667fafc20ad216f728f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x-2y+C=0 \\ \\xrightarrow{x=-4 \\ ; \\ y=3} \\ 6\\cdot (-4)-2\\cdot 3+C=0\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"414\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06deb5d0e5a9ede569d5df8ebb81efac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-24-6+C=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"129\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9f169aa941fe39d794dc14e328e6dcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-30+C=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"99\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-98e55b016b593186a4639d6755ce98be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C=30\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Die implizite, allgemeine oder kartesische Gleichung der Geraden lautet also:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df38e3e4991749df96b24d202e033f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{6x-2y+30=0}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"466\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt k\u00f6nnen wir die explizite Gleichung der Geradenl\u00f6sung f\u00fcr die Unbekannte bestimmen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> der impliziten Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6418dd689e7fe9e034c7bc979d6b3401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x-2y+30=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7bea9f5614281ca073dd0a12624dd8aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2y=-6x-30\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c61c7e6c7767bff1a07380b2aab05ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-6x-30}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7286e57d0abbd99815b3324e8194227c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{y=3x+15}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"427\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daher ist die Steigung der Geraden gleich 3 (Term vor der unabh\u00e4ngigen Variablen).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> ).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-260107cba86a7b21e919180b1130050e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und aus dem Wert der Steigung der Geraden k\u00f6nnen wir die Punkt-Steigungsgleichung der Geraden mit ihrer Formel berechnen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f0c8bfe8364c4962a61ff66ab943aeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-P_2=m(x-P_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5a35b2e6a22317e902b5cce4815ccd6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-3=3(x-(-4))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-671def0bdf4e7cf62d2019cc5187a130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{y-3=3(x+4)}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"462\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 3<\/h3>\n<p> Bestimmen Sie 3 Punkte auf der folgenden Linie, ausgedr\u00fcckt als implizite oder allgemeine Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92f98258fe0de7bfdabef5dfa0b9678c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x+2y-8 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"122\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Um einen Punkt auf einer Linie zu berechnen, m\u00fcssen wir lediglich einer der Variablen einen Wert zuweisen und dann den Wert der anderen Variablen an diesem Punkt ermitteln.<\/p>\n<p class=\"has-text-align-left\"> Wir berechnen einen ersten Punkt indem wir tun <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d762821a7c6da83f02380639f43ef8fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21bfe654b6dd49f110abf58c6d3df214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot 0+2y-8 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"134\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe00aaf21ee98c3036532591e2796987_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cfff981f9d77c69e6b8339ecd141562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = \\cfrac{8}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"44\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07c6b260c2e43f4545a7c1974de73cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03dff5e2c9b5389babf595bd961ba962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P_1(0,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dann finden wir einen zweiten Punkt, der der Variablen einen anderen Wert gibt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<p> Zum Beispiel <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-adfd4b59a1c96b58188448b5fe50dec7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c99451893ed7c2a58fb423eeddcf5258_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot 1+2y-8 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"134\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4419d32ace4c25320f6875b7acd275f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y = 8-4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b72eeccbfe9aa758d0cbda671640ad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f9787827f5407a73e53398776daff63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = \\cfrac{4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"44\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e47e82982611531964ade31826b9e254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b797693671aff6ba5e46c9808a3f20d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P_2(1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich berechnen wir durch L\u00f6sen einen dritten Punkt <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6aff213e5b8cce8e689840fc8f6b8413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41842e99d77e585e9c7a5417f41a3167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot 2+2y-8 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"134\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-376f7dee73fe357a8c79a157c8b4a966_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y = 8-8\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce9418065ff9df3f3d58e578cb45a9b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fff4ce6b62556e0291e3d191a0d05ee9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = \\cfrac{0}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"44\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0d3e85f938e2ecfcc840836d5698d72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-930774e05056e449ca78c16287f4481c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P_3(2,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 4<\/h3>\n<p> Finden Sie alle Gleichungen der durch den Punkt definierten Linie<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> und der Vektor <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7fe319cb0fecedf2052e6c1e4c856733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e319f5d0c3f211308f1489efbc6665d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,4) \\qquad \\qquad \\vv{\\text{v}}=(-3,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"223\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zun\u00e4chst finden wir die Vektorgleichung der Geraden aus ihrer Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16e43c9d65f7fae5b0e20a3caca1df38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y)=(P_1,P_2)+t\\cdot (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49afd56a168dbacfe6fa06f284c36b95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{(x,y)=(-1,4)+t\\cdot (-3,6)} \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"547\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zweitens finden wir die parametrischen Gleichungen der Geraden durch die entsprechende Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2e6878c4d9b80337639f5fa7728a9f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x=P_1+t\\cdot\\text{v}_1 \\\\[1.7ex] y=P_2+t\\cdot\\text{v}_2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3bf46da9a68147118874a619f918077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\begin{cases} \\bm{x=-1-3t} \\\\[1.7ex] \\bm{y=4+6t} \\end{cases} \\quad \\vphantom{\\cfrac{\\cfrac{1}{2}}{\\cfrac{1}{2}}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"468\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und wir bestimmen auch die kontinuierliche Gleichung der Geraden anhand ihrer Formel: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-913e6797e350e331ce17df6b5c074f91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-P_1}{\\text{v}_1}=\\cfrac{y-P_2}{\\text{v}_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2712f4cc5c10283e89aab8c241bd0c6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-(-1)}{-3}=\\cfrac{y-4}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"134\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-507d9c42afbe40a47426d930cd655acd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\cfrac{\\bm{x+1}}{\\bm{-3}}\\bm{=}\\cfrac{\\bm{y-4}}{\\bm{6}}\\quad \\vphantom{\\Biggl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"434\" style=\"vertical-align: -28px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kommen wir nun dazu, die implizite oder allgemeine Gleichung der Geraden zu finden. Dazu kreuzen wir die beiden Br\u00fcche der stetigen Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e622246f6aee3aeeae4e46c2ab273448_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot (x+1)= -3 \\cdot (y-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-233758c9cfa6e67037f0d9ac3491646b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x+6= -3y+12\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"144\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dce0a8f03df8900f43824c6dfe4965db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x+6+3y-12=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"161\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca892beb6a3dad6875963c37005a06ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{6x+3y-6=0}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"457\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt k\u00f6nnen wir die explizite Gleichung der Geradenl\u00f6sung f\u00fcr die Unbekannte bestimmen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> der impliziten Gleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db879a59b656865b385242dd4c936671_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x+3y-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"122\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa0f4f461dead11823094f6c10e35131_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3y=-6x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce147ff77e418ccac9df420b80b7955f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-6x+6}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b069b281c719aacd05068d3e41d953e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad \\bm{y=-2x+2}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"432\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daher entspricht die Steigung der Linie -2 (Term, der die unabh\u00e4ngige Variable begleitet).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> ).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4cb69f8df8ea8c5935576ece37a640c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und damit k\u00f6nnen wir die Punkt-Steigungsgleichung der Geraden mit ihrer Formel berechnen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f0c8bfe8364c4962a61ff66ab943aeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-P_2=m(x-P_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f6be33c2f75c28cefd5edacf7415db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4=-2(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d31a0383074e6488a39acdf48b73cc64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\bm{y-4=-2(x+1)}\\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"476\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Um schlie\u00dflich die Segmentgleichung der Geraden zu finden, berechnen wir die Schnittpunkte der Geraden mit den Achsen OX und OY und verwenden dann ihre Formel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9ee9a398efc0528db8f6e51a774b2116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-56\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Schnittpunkt mit der Abszissenachse (X-Achse)<\/span> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b3265e1ea6c0695580197ddb6f267c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0=-2x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"95\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-78019a5b7c3b3aed3f574c422cf48ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e05ddb74466e215ecc9c0b5dbe90e54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{2}{2} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad6e487184338b18ddb30720fb02a024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(1, 0\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Schnittpunkt mit der y-Achse (Y-Achse)<\/span> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbbba3968c6e6c8f20894507d3feed33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2\\cdot 0+2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-552d8ed773e160e229551b39aff39445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5350b2cb3b61a50c3ecb754aa4c44518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(0,2\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2f3b0119758235dab9c6000508936ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{a}+\\cfrac{y}{b} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0a9eb3e730af50fc6c543600ca010ce7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{exemple}{HTML}{2196F3} \\color{exemple} \\boxed{ \\color{black} \\quad\\cfrac{\\bm{x}}{\\bm{1}}+\\cfrac{\\bm{y}}{\\bm{2}} \\bm{= 1} \\quad \\vphantom{\\Biggl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"408\" style=\"vertical-align: -28px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 5<\/h3>\n<p> Finden Sie die Gleichung der Geraden, die durch die folgenden zwei Punkte verl\u00e4uft: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e2b3beb2ff2f058df5534f1bd6b925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1 (4,-1) \\qquad \\qquad P_2(5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"201\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Da wir bereits zwei Punkte auf der Geraden kennen, wenden wir die Formel f\u00fcr die Geradengleichung direkt auf 2 gegebene Punkte an:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f97d63f216910e7979937859fb90a10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_1= \\cfrac{y_2-y_1}{x_2-x_1} (x-x_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"193\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt setzen wir die kartesischen Koordinaten der Punkte in die Formel ein:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c75c3e6820d69b59b21ff79e2aee3055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-(-1)= \\cfrac{2-(-1)}{5-4} (x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"214\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich berechnen wir die Steigung der Geraden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3646c71be87d906417e00a512450ca9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y+1= \\cfrac{3}{1} (x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"128\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a70434a562eb7d94f6ff02d23de896a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y+1= 3(x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Die Gleichung der Geraden, die durch diese beiden Punkte geht, lautet daher:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e7c7f608e0e1ba52d453cad7a29e99d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y+1= 3(x-4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hier finden Sie die Formeln f\u00fcr alle Arten von Geradengleichungen. Dar\u00fcber hinaus k\u00f6nnen Sie Beispiele f\u00fcr deren Berechnung sehen und zus\u00e4tzlich mit gel\u00f6sten \u00dcbungen zu den Geradengleichungen \u00fcben. Wie lauten alle Gleichungen der Geraden? Denken Sie daran, dass die mathematische Definition einer Linie eine Menge aufeinanderfolgender Punkte ist, die in derselben Richtung ohne Kurven oder &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/de\/liniengleichungen-alle-formeln-beispiele-geloste-ubungen\/\"> <span class=\"screen-reader-text\">Liniengleichungen<\/span> Weiterlesen &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[16],"tags":[],"class_list":["post-264","post","type-post","status-publish","format-standard","hentry","category-polynome"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Liniengleichungen -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/de\/liniengleichungen-alle-formeln-beispiele-geloste-ubungen\/\" \/>\n<meta property=\"og:locale\" content=\"de_DE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Liniengleichungen -\" \/>\n<meta property=\"og:description\" content=\"Hier finden Sie die Formeln f\u00fcr alle Arten von Geradengleichungen. 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