{"id":265,"date":"2023-07-10T06:09:02","date_gmt":"2023-07-10T06:09:02","guid":{"rendered":"https:\/\/mathority.org\/it\/equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti\/"},"modified":"2023-07-10T06:09:02","modified_gmt":"2023-07-10T06:09:02","slug":"equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti\/","title":{"rendered":"Equazioni di linea"},"content":{"rendered":"<p>Qui troverai le formule per tutti i tipi di equazioni della retta. Inoltre, potrai vedere esempi di come vengono calcolati e, inoltre, esercitarti con esercizi risolti delle equazioni della retta. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuales-son-todas-las-ecuaciones-de-la-recta\"><\/span> Quali sono tutte le equazioni della retta?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ricorda che la definizione matematica di linea \u00e8 un insieme di punti consecutivi rappresentati nella stessa direzione senza curve o angoli.<\/p>\n<p> Quindi, per esprimere analiticamente una qualsiasi retta nel piano (in R2) utilizziamo le equazioni della retta, e per trovarle basta un punto appartenente alla retta e il vettore direzione di detta retta. Solo con questi due elementi geometrici si possono trovare assolutamente tutte le diverse equazioni della retta, che sono le seguenti:<\/p>\n<p> <strong>Le equazioni della retta sono l&#8217;equazione vettoriale, le equazioni parametriche, l&#8217;equazione continua, l&#8217;equazione implicita (o generale), l&#8217;equazione esplicita, l&#8217;equazione punto-pendenza e l&#8217;equazione canonica (o segmentale).<\/strong><\/p>\n<p> Tutti i tipi di equazioni di linea hanno lo stesso obiettivo: rappresentare matematicamente una linea. Ma ogni equazione della retta ha le sue propriet\u00e0 e quindi, a seconda del problema, \u00e8 meglio utilizzare l&#8217;una o l&#8217;altra. <\/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=\"equazioni di una linea pdf\" width=\"287\" height=\"273\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Una volta visto il concetto di equazioni di linea, passiamo ora ad analizzare le caratteristiche di ciascun tipo di equazione di linea in particolare. Di seguito hai una spiegazione dettagliata dei diversi tipi di equazioni della riga, ma se vuoi puoi andare direttamente alla fine della <a href=\"https:\/\/mathority.org\/it\/equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti\/\">tabella riassuntiva con le formule di tutte le equazioni della riga<\/a> . <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-vectorial-de-la-recta\"><\/span> Equazione vettoriale della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> S\u00ec<\/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> \u00e8 il vettore direzione della linea e<\/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> un punto che appartiene a destra:<\/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> La <strong>formula per l&#8217;equazione vettoriale della retta<\/strong> \u00e8:<\/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> Oro:<\/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> E<\/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> sono le coordinate cartesiane di ogni punto della retta.<\/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> E<\/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> sono le coordinate di un punto noto facente parte della linea<\/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> E<\/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> sono le componenti del vettore direzione della linea<\/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> \u00e8 uno scalare (un numero reale) il cui valore dipende da ciascun punto della linea.<\/li>\n<\/ul>\n<p> \u00c8 l&#8217;equazione vettoriale della retta nel piano, cio\u00e8 quando si lavora con punti e vettori di 2 coordinate (in R2). Tuttavia, se eseguissimo i calcoli nello spazio (in R3), dovremmo aggiungere un&#8217;ulteriore componente all&#8217;equazione della retta: <\/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> Equazioni parametriche della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Le equazioni parametriche di una linea possono essere ottenute dalla sua equazione vettoriale:<\/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> Per prima cosa moltiplichiamo il parametro<\/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> dal vettore direzione di destra:<\/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> Successivamente, aggiungiamo le coordinate X e Y:<\/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> E, infine, cancellando ciascuna variabile separatamente, otteniamo le equazioni parametriche della linea:<\/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> Oro:<\/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> E<\/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> sono le coordinate cartesiane di ogni punto della retta.<\/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> E<\/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> sono le coordinate di un punto noto facente parte della linea<\/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> E<\/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> sono le componenti del vettore direzione della linea<\/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> \u00e8 uno scalare (un numero reale) il cui valore dipende da ciascun punto della linea.<\/li>\n<\/ul>\n<p> Come prima, queste sono le equazioni parametriche della retta nel piano (in R2), ma per trovare le equazioni parametriche della retta nello spazio (in R3) sarebbe necessario aggiungere un&#8217;altra equazione per la terza variabile Z: <\/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>Equazione continua della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> L&#8217;equazione continua di qualsiasi linea pu\u00f2 essere dedotta dalle sue equazioni parametriche:<\/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> Se cancelliamo l&#8217;impostazione<\/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> da ciascuna equazione parametrica si ottengono le seguenti espressioni:<\/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 Uguagliando le due equazioni risultanti, otteniamo l&#8217;equazione continua della retta:<\/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> In breve, l\u2019 <strong>equazione continua della retta<\/strong> \u00e8:<\/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> Oro:<\/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> E<\/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> sono le coordinate cartesiane di ogni punto della retta.<\/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> E<\/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> sono le coordinate di un punto noto facente parte della linea<\/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> E<\/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> sono le componenti del vettore direzione della linea<\/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> Questa formula serve per l&#8217;equazione continua della linea quando si lavora in 2 dimensioni (in 2D). Ma se eseguissimo operazioni in 3 dimensioni (3D), dovremmo aggiungere un componente aggiuntivo all&#8217;equazione della linea: <\/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> Equazione implicita o generale della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> S\u00ec<\/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> \u00e8 il vettore direzione della linea e<\/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> un punto che appartiene a destra:<\/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> La formula per l&#8217; <strong>equazione implicita, generale o cartesiana della retta<\/strong> \u00e8:<\/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> Oro:<\/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> E<\/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> sono le coordinate cartesiane di ogni punto della retta.<\/li>\n<li> il coefficiente\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> \u00e8 la seconda componente del vettore direzione della retta:<\/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> il coefficiente\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> \u00e8 la prima componente del vettore di direzione cambiato segno:<\/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> il coefficiente\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> viene calcolato sostituendo il punto noto<\/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> nell&#8217;equazione della retta.<\/li>\n<\/ul>\n<p> la formula, l&#8217;equazione implicita di una retta si pu\u00f2 ottenere anche moltiplicando le frazioni dell&#8217;equazione continua. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-explicita-de-la-recta\"><\/span> Equazione esplicita della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> La formula per l&#8217; <strong>equazione esplicita della retta<\/strong> \u00e8:<\/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> Oro:<\/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> \u00e8 la pendenza della retta.<\/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> la sua intercetta y, cio\u00e8 l&#8217;altezza alla quale interseca l&#8217;asse Y.<\/li>\n<\/ul>\n<p> Nella sezione seguente vedrai come vengono determinati i parametri<\/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> E<\/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> della retta Ma, in particolare, un altro modo per trovare l&#8217;equazione esplicita \u00e8 utilizzare l&#8217;equazione implicita; per questo, l&#8217;incognita deve essere risolta<\/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> dell&#8217;equazione implicita.<\/p>\n<h4 class=\"wp-block-heading\"> Significato dei parametri m e n<\/h4>\n<p> Come abbiamo visto nella definizione dell&#8217;equazione esplicita della retta, il parametro<\/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> \u00e8 la pendenza della retta e<\/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> la sua intercetta y. Ma cosa significa? Vediamolo dalla rappresentazione grafica di una linea: <\/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=\"Qual \u00e8 l'equazione esplicita della retta y=mx+b\" class=\"wp-image-1455\" width=\"339\" height=\"339\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Il termine indipendente<\/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>\u00e8 il punto di intersezione della linea con l&#8217;asse del computer<\/strong> (asse OY). Ad esempio, nel grafico sopra<\/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> \u00e8 uguale a 1 perch\u00e9 la linea interseca l&#8217;asse y in y=1.<\/p>\n<p> D&#8217;altra parte, il termine<\/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>indica la pendenza della retta<\/strong> , cio\u00e8 la sua inclinazione. Come vedi nel grafico,<\/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> \u00e8 uguale a 2 poich\u00e9 la linea sale di 2 unit\u00e0 verticali per 1 unit\u00e0 orizzontale.<\/p>\n<p> Ovviamente se la pendenza \u00e8 positiva la funzione \u00e8 crescente (sale), invece se la pendenza \u00e8 negativa la funzione \u00e8 decrescente (scende).<\/p>\n<h5 class=\"wp-block-heading\"> Calcolare la pendenza di una retta<\/h5>\n<p> Una volta che sappiamo esattamente qual \u00e8 la pendenza di una retta, vediamo come viene calcolata. Pertanto, ci sono 3 modi diversi per determinare numericamente la pendenza di una linea:<\/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;\">Dati due punti diversi sulla retta\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> E<\/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> La pendenza della retta \u00e8 pari a:<\/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;\">S\u00ec\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> \u00e8 il vettore direzione della retta, la sua pendenza \u00e8:<\/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;\">S\u00ec\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> \u00e8 l&#8217;angolo formato dalla retta con l&#8217;asse delle ascisse (asse X), la pendenza della retta \u00e8 equivalente alla tangente di detto angolo: <\/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=\"formula per l'equazione esplicita della retta\" 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> Equazione punto-pendenza della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> La formula per l&#8217; <strong>equazione punto-pendenza della retta<\/strong> \u00e8:<\/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> Oro:<\/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> \u00e8 la pendenza della retta.<\/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> sono le coordinate di un punto sulla retta <\/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> Equazione canonica o segmentale della retta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Sebbene questa variante dell&#8217;equazione della retta sia meno conosciuta, l&#8217;equazione canonica della retta si pu\u00f2 ottenere dai punti di intersezione della retta con gli assi cartesiani.<\/p>\n<p> Siano i due punti di intersezione con gli assi di una data retta:<\/p>\n<p class=\"has-text-align-center\"> Tagliare con l&#8217;asse X:<\/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\"> Taglio con asse Y:<\/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> La <strong>formula per l&#8217;equazione canonica della retta<\/strong> \u00e8: <\/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=\"equazioni del calcolatore di linea\" class=\"wp-image-3261\" width=\"297\" height=\"298\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> In matematica, l&#8217;equazione canonica della retta \u00e8 anche chiamata equazione segmentale o equazione simmetrica.<\/p>\n<p> D&#8217;altra parte, i coefficienti<\/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> E<\/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> Possono anche essere trovati dall&#8217;equazione generale della retta utilizzando le seguenti formule: <\/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> Tutte le equazioni della retta (formule)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Riassumendo, ecco una tabella in cui sono riportate le formule di tutte le equazioni della retta: <\/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> Esempio di calcolo delle equazioni della retta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ora che abbiamo visto tutta la spiegazione dell&#8217;equazione della retta, vediamo come si risolve un tipico problema delle equazioni della retta:<\/p>\n<ul>\n<li> Trova tutte le equazioni della retta determinata dal punto\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> e il vettore<\/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> Innanzitutto troviamo l&#8217;equazione vettoriale della retta dalla sua formula:<\/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> Sostituisci semplicemente le coordinate del punto e il vettore nella formula:<\/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> In secondo luogo, troviamo le equazioni parametriche della retta attraverso la sua formula corrispondente:<\/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> E determiniamo anche l&#8217;equazione continua della retta con la sua formula: <\/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> Come hai visto, le equazioni vettoriali, parametriche e continue sono facili da calcolare, basta utilizzare le rispettive formule.<\/p>\n<p> Passiamo ora alla ricerca dell&#8217;equazione generale (o implicita) della retta. Per fare ci\u00f2, incrociamo le due frazioni dell&#8217;equazione continua: <\/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> Ora possiamo determinare l&#8217;equazione esplicita della retta risolvendo l&#8217;incognita<\/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> dell&#8217;equazione implicita: <\/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> Pertanto la pendenza della retta \u00e8 pari a 2 (termine che accompagna la variabile indipendente<\/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> E con questo possiamo calcolare l&#8217;equazione punto-pendenza della retta con la sua formula: <\/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> Infine, per trovare l&#8217;equazione segmentale della retta calcoliamo i suoi punti di intersezione con gli assi OX e OY quindi applichiamo la sua formula:<\/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;\">Punto di intersezione con l&#8217;asse delle ascisse (asse X)<\/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;\">Punto di intersezione con l&#8217;asse y (asse Y)<\/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> Equazione della retta passante per due punti<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Un altro problema molto comune nelle equazioni delle rette \u00e8 trovare l&#8217;equazione della retta determinata da due punti dati. Sebbene possiamo calcolare il vettore direzione della linea con i 2 punti e quindi l&#8217;equazione, di seguito ti forniamo una formula con la quale puoi trovare direttamente e facilmente l&#8217;equazione di detta linea.<\/p>\n<p> Considera due punti situati su una linea:<\/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> La <strong>formula per trovare l&#8217;equazione della retta dai suoi 2 punti<\/strong> \u00e8:<\/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> Questa formula ci permette di calcolare direttamente l&#8217;equazione punto-pendenza della retta quando ci vengono dati 2 punti attraverso i quali passa la retta. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-las-ecuaciones-de-la-recta\"><\/span> Risolti problemi di equazioni della retta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Trova l&#8217;equazione vettoriale, le equazioni parametriche e l&#8217;equazione continua della linea definita dal punto<\/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> e il suo vettore direttivo<\/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> Sii entrambi: <\/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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Innanzitutto, calcoliamo l&#8217;equazione vettoriale della retta dalla sua formula: <\/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\"> Troviamo quindi le equazioni parametriche della retta utilizzando la formula corrispondente: <\/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\"> E, infine, determiniamo l&#8217;equazione continua della retta con la rispettiva formula: <\/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\">Esercizio 2<\/h3>\n<p> Trova l&#8217;equazione implicita, l&#8217;equazione esplicita e l&#8217;equazione punto-pendenza della linea determinata dal punto<\/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> e il suo vettore di direzione \u00e8 <\/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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La formula per l&#8217;equazione implicita della retta \u00e8:<\/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\"> Dobbiamo quindi trovare i coefficienti A, B e C. Le incognite A e B si ottengono dalle coordinate del vettore direzione della retta, perch\u00e9 \u00e8 sempre verificata la seguente uguaglianza:<\/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\"> Di conseguenza, il coefficiente A \u00e8 la seconda coordinata del vettore, e il coefficiente B \u00e8 la prima coordinata del vettore cambiato segno:<\/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\"> Dobbiamo quindi trovare solo il coefficiente C. Per fare ci\u00f2, dobbiamo sostituire nella sua equazione il punto che sappiamo appartenere alla retta: <\/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\"> Quindi l\u2019equazione implicita, generale o cartesiana della retta \u00e8:<\/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\"> Ora possiamo determinare l&#8217;equazione esplicita della retta risolvendo l&#8217;incognita<\/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> dell&#8217;equazione implicita: <\/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\"> Pertanto la pendenza della retta \u00e8 pari a 3 (termine prima della variabile indipendente<\/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\"> E, dal valore della pendenza della retta, possiamo calcolare l&#8217;equazione punto-pendenza della retta con la sua formula: <\/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\">Esercizio 3<\/h3>\n<p> Determina 3 punti sulla riga seguente, espressi come equazione implicita o generale: <\/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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per calcolare un punto su una linea, dobbiamo semplicemente assegnare un valore a una delle variabili e poi trovare il valore dell&#8217;altra variabile in quel punto.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo un primo punto facendo <\/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\"> Troviamo quindi un secondo punto che d\u00e0 un altro valore alla variabile<\/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> Per esempio <\/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\"> E, infine, calcoliamo un terzo punto risolvendo <\/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\">Esercizio 4<\/h3>\n<p> Trova tutte le equazioni della retta definita dal punto<\/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> e il vettore <\/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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Innanzitutto troviamo l&#8217;equazione vettoriale della retta dalla sua formula: <\/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\"> In secondo luogo, troviamo le equazioni parametriche della retta attraverso la sua formula corrispondente: <\/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\"> E determiniamo anche l&#8217;equazione continua della linea usando la sua formula: <\/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\"> Passiamo ora a trovare l&#8217;equazione implicita o generale della retta. Per fare ci\u00f2, incrociamo le due frazioni dell&#8217;equazione continua: <\/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\"> Ora possiamo determinare l&#8217;equazione esplicita della retta risolvendo l&#8217;incognita<\/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> dell&#8217;equazione implicita: <\/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\"> Pertanto la pendenza della retta equivale a -2 (termine che accompagna la variabile indipendente<\/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\"> E con questo possiamo calcolare l&#8217;equazione punto-pendenza della retta con la sua formula: <\/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\"> Infine, per trovare l&#8217;equazione segmentale della retta, calcoliamo i punti di intersezione della retta con gli assi OX e OY quindi utilizziamo la sua formula:<\/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;\">Punto di intersezione con l&#8217;asse delle ascisse (asse X)<\/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;\">Punto di intersezione con l&#8217;asse y (asse Y)<\/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\">Esercizio 5<\/h3>\n<p> Trova l&#8217;equazione della retta che passa per i seguenti due punti: <\/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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Poich\u00e9 conosciamo gi\u00e0 due punti sulla retta, applichiamo direttamente la formula per l&#8217;equazione della retta a 2 punti dati:<\/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\"> Ora sostituiamo le coordinate cartesiane dei punti nella formula:<\/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\"> E, infine, calcoliamo la pendenza della retta: <\/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\"> L&#8217;equazione della retta che passa per questi due punti \u00e8 quindi:<\/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>Qui troverai le formule per tutti i tipi di equazioni della retta. Inoltre, potrai vedere esempi di come vengono calcolati e, inoltre, esercitarti con esercizi risolti delle equazioni della retta. Quali sono tutte le equazioni della retta? Ricorda che la definizione matematica di linea \u00e8 un insieme di punti consecutivi rappresentati nella stessa direzione senza &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti\/\"> <span class=\"screen-reader-text\">Equazioni di linea<\/span> Leggi altro &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-265","post","type-post","status-publish","format-standard","hentry","category-polinomi"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Equazioni di linea -<\/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\/it\/equazioni-di-linea-tutte-le-formule-esempi-esercizi-risolti\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equazioni di linea -\" \/>\n<meta property=\"og:description\" content=\"Qui troverai le formule per tutti i tipi di equazioni della retta. Inoltre, potrai vedere esempi di come vengono calcolati e, inoltre, esercitarti con esercizi risolti delle equazioni della retta. Quali sono tutte le equazioni della retta? 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