{"id":77643,"date":"2024-09-23T09:00:00","date_gmt":"2024-09-23T12:00:00","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=77643"},"modified":"2024-10-15T09:54:04","modified_gmt":"2024-10-15T12:54:04","slug":"entenda-a-equacao-geral-da-reta-na-geometria-analitica","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/","title":{"rendered":"Equa\u00e7\u00e3o geral da reta na Geometria Anal\u00edtica: o que \u00e9, componentes e mais!"},"content":{"rendered":"\n<p>A Geometria Anal\u00edtica \u00e9 a ferramenta para traduzir o comportamento das figuras geom\u00e9tricas, especialmente quando representadas no plano cartesiano. Entre os conceitos mais fundamentais est\u00e1 a equa\u00e7\u00e3o geral da reta, que permite descrever algebricamente qualquer reta por meio das coordenadas <span class=\"latex-block\"><span class=\"latex-inline\">$$x e y$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<p>Compreender como encontrar essa equa\u00e7\u00e3o a partir de dois pontos conhecidos \u00e9 crucial para resolver quest\u00f5es de Geometria Anal\u00edtica. A utiliza\u00e7\u00e3o do determinante para obter a equa\u00e7\u00e3o geral da reta \u00e9 um passo a passo indispens\u00e1vel para garantir precis\u00e3o e agilidade nas resolu\u00e7\u00f5es.<\/p>\n\n\n\n<p>Vamos explorar agora exemplos pr\u00e1ticos e exerc\u00edcios que demonstram a aplica\u00e7\u00e3o dessa teoria por meio desse artigo preparado pelo Estrat\u00e9gia Militares para voc\u00ea. Entenda tudo sobre a equa\u00e7\u00e3o geral da reta a seguir!<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado\t<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/#O-que-e-a-equacao-geral-da-reta\" >O que \u00e9 a equa\u00e7\u00e3o geral da reta<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/#Passos-para-encontrar-a-equacao-geral-da-reta\" >Passos para encontrar a equa\u00e7\u00e3o geral da reta<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/#Exemplos-praticos-e-resolucao-de-exercicios\" >Exemplos pr\u00e1ticos e resolu\u00e7\u00e3o de exerc\u00edcios<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"o-que-\u00e9-a-equa\u00e7\u00e3o-geral-da-reta\"><span class=\"ez-toc-section\" id=\"O-que-e-a-equacao-geral-da-reta\"><\/span>O que \u00e9 a equa\u00e7\u00e3o geral da reta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>A equa\u00e7\u00e3o geral da reta \u00e9 uma forma de representar, de maneira alg\u00e9brica, a reta no plano cartesiano. Dada por <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$ax + by + c = 0$<\/span><span class=\"latex-inline\">$<\/span>, essa equa\u00e7\u00e3o traduz a posi\u00e7\u00e3o e a inclina\u00e7\u00e3o de uma reta atrav\u00e9s das constantes <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$a$<\/span><span class=\"latex-inline\">$<\/span>, <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$b$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$c$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<p>Esse \u00e9 um conceito fundamental na Geometria Anal\u00edtica, permitindo a descri\u00e7\u00e3o precisa e matem\u00e1tica dos comportamentos de retas.<\/p>\n\n\n\n<p>A import\u00e2ncia dessa equa\u00e7\u00e3o est\u00e1 em sua capacidade de generalizar a representa\u00e7\u00e3o das diversas poss\u00edveis inclina\u00e7\u00f5es e posi\u00e7\u00f5es das retas no plano cartesiano.<\/p>\n\n\n\n<p>Seja uma reta inclinada, horizontal ou vertical, a equa\u00e7\u00e3o geral consegue abrang\u00ea-las todas. Tal versatilidade a torna uma ferramenta essencial tanto em c\u00e1lculos te\u00f3ricos quanto em aplica\u00e7\u00f5es pr\u00e1ticas.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"componentes-da-equa\u00e7\u00e3o\">Componentes da equa\u00e7\u00e3o<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$<strong>ax<\/strong>$<\/span><span class=\"latex-inline\">$<\/span>: O termo <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$a$<\/span><span class=\"latex-inline\">$<\/span>est\u00e1 associado ao coeficiente de <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$x$<\/span><span class=\"latex-inline\">$<\/span> e influencia a inclina\u00e7\u00e3o da reta. Quanto maior for o valor absoluto de <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$a$<\/span><span class=\"latex-inline\">$<\/span>, mais inclinada ser\u00e1 a reta.<\/li>\n\n\n\n<li><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$<strong>by<\/strong>$<\/span><span class=\"latex-inline\">$<\/span>: O coeficiente <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$b$<\/span><span class=\"latex-inline\">$<\/span> multiplica a coordenada<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y$<\/span><span class=\"latex-inline\">$<\/span> e tamb\u00e9m contribui para a inclina\u00e7\u00e3o da reta.<\/li>\n\n\n\n<li><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$<strong>c<\/strong>$<\/span><span class=\"latex-inline\">$<\/span>: Termo constante que desloca a reta verticalmente ou horizontalmente no plano.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"propriedades-e-caracter\u00edsticas\">Propriedades e caracter\u00edsticas<\/h3>\n\n\n\n<p>Ao modificar os valores de a, b, e c, \u00e9 poss\u00edvel alterar a inclina\u00e7\u00e3o e a posi\u00e7\u00e3o da reta no plano. Por exemplo:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Se <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$a = 0$<\/span><span class=\"latex-inline\">$<\/span>, a reta \u00e9 horizontal.<\/li>\n\n\n\n<li>Se <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$b = 0$<\/span><span class=\"latex-inline\">$<\/span>, a reta \u00e9 vertical.<\/li>\n\n\n\n<li>Se tanto <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$a$<\/span><span class=\"latex-inline\">$<\/span> quanto <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$b$<\/span><span class=\"latex-inline\">$<\/span> s\u00e3o diferentes de zero, a reta ser\u00e1 inclinada.<\/li>\n<\/ul>\n\n\n\n<p>Por meio dessas propriedades, a equa\u00e7\u00e3o geral da reta facilita a an\u00e1lise comparativa e o estudo de suas interse\u00e7\u00f5es com outras figuras geom\u00e9tricas no plano cartesiano.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exemplo-visual-e-explicativo\">Exemplo visual e explicativo:<\/h3>\n\n\n\n<p>Para ilustrar, considere a equa\u00e7\u00e3o <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$2x + 3y &#8211; 6 = 0$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<p>Ao reescrevermos essa equa\u00e7\u00e3o de forma isolada para <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y$<\/span><span class=\"latex-inline\">$<\/span>, obtemos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y = -2\/3x + 2$<\/span><span class=\"latex-inline\">$<\/span>. A inclina\u00e7\u00e3o (coeficiente angular) \u00e9 <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$-2\/3$<\/span><span class=\"latex-inline\">$<\/span>, indicando que a reta desce dois unidades no eixo <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y$<\/span><span class=\"latex-inline\">$<\/span> para cada tr\u00eas unidades que avan\u00e7a no eixo <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$x$<\/span><span class=\"latex-inline\">$<\/span>. O intercepto <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y$<\/span><span class=\"latex-inline\">$<\/span> (onde a reta cruza o eixo <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$y$<\/span><span class=\"latex-inline\">$<\/span>) \u00e9 <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$2$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<p>Tais an\u00e1lises s\u00e3o \u00fateis em diversas situa\u00e7\u00f5es, desde problemas simples de localiza\u00e7\u00e3o de pontos at\u00e9 aplica\u00e7\u00f5es mais complexas em f\u00edsica e engenharia. A compreens\u00e3o e a utiliza\u00e7\u00e3o eficiente da equa\u00e7\u00e3o geral da reta constituem alicerces indispens\u00e1veis para quem estuda ou trabalha com matem\u00e1tica e suas aplica\u00e7\u00f5es.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"passos-para-encontrar-a-equa\u00e7\u00e3o-geral-da-reta\"><span class=\"ez-toc-section\" id=\"Passos-para-encontrar-a-equacao-geral-da-reta\"><\/span>Passos para encontrar a equa\u00e7\u00e3o geral da reta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Dominar a constru\u00e7\u00e3o da equa\u00e7\u00e3o geral da reta \u00e9 essencial. A seguir, descreveremos de forma detalhada os passos para encontrar essa equa\u00e7\u00e3o, utilizando exemplos pr\u00e1ticos para melhor compreens\u00e3o.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"passo-1-identificar-as-coordenadas-dos-pontos\">Passo 1: Identificar as coordenadas dos pontos<\/h3>\n\n\n\n<p>Para come\u00e7ar, precisamos conhecer as coordenadas de dois pontos distintos que pertencem \u00e0 reta. Suponhamos que temos dois pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A(xA, yA)$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B(xB, yB)$<\/span><span class=\"latex-inline\">$<\/span>. \u00c9 a partir deles que calcularemos o determinante necess\u00e1rio para encontrar a equa\u00e7\u00e3o da reta.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"passo-2-construir-a-matriz-dos-pontos\">Passo 2: Construir a matriz dos pontos<\/h3>\n\n\n\n<p>Com os pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B$<\/span><span class=\"latex-inline\">$<\/span> definidos, constru\u00edmos uma matriz <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$3&#215;3$<\/span><span class=\"latex-inline\">$<\/span> que inclui as coordenadas desses pontos. A matriz ter\u00e1 a seguinte forma:<\/p>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\begin{vmatrix}\nxA &#038; xB &#038; x \\\nyA &#038; yB &#038; y \\\n1 &#038; 1 &#038; 1\n\\end{vmatrix}\n= 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<p>Esse determinante deve ser calculado e igualado a zero para prosseguirmos.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"passo-3-calcular-o-determinante\">Passo 3: Calcular o determinante<\/h3>\n\n\n\n<p>Vamos calcular o determinante da matriz definida no passo anterior. Para ilustrar, considere os pontos A(2, 4) e B(3, 7). A matriz correspondente \u00e9:<\/p>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\begin{vmatrix}\n2 &#038; 3 &#038; x \\\n4 &#038; 7 &#038; y \\\n1 &#038; 1 &#038; 1\n\\end{vmatrix}\n= 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<p>Multiplicamos e somamos os produtos das diagonais:<\/p>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n2 \\cdot 7 \\cdot 1 + 4 \\cdot 1 \\cdot x + 1 \\cdot 3 \\cdot y &#8211; (1 \\cdot 7 \\cdot x + 2 \\cdot 1 \\cdot y + 4 \\cdot 3 \\cdot 1) = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"passo-4-simplificar-a-equa\u00e7\u00e3o\">Passo 4: Simplificar a equa\u00e7\u00e3o<\/h3>\n\n\n\n<p>Ap\u00f3s realizar as multiplica\u00e7\u00f5es e somas descritas no passo anterior, obtemos:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[14 + 4x + 3y &#8211; 7x &#8211; 2y &#8211; 12 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>Simplificamos essa express\u00e3o para encontrar a equa\u00e7\u00e3o geral da reta. Neste caso, reorganizando os termos, obtemos:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + y + 2 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exemplos-pr\u00e1ticos\">Exemplos pr\u00e1ticos<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"exemplo-1\">Exemplo 1<\/h4>\n\n\n\n<p>Encontre a equa\u00e7\u00e3o da reta que passa pelos pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A(2, 1)$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B(4, 7)$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\begin{vmatrix}\n2 &#038; 4 &#038; x \\\n1 &#038; 7 &#038; y \\\n1 &#038; 1 &#038; 1\n\\end{vmatrix}\n= 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<p>Calculando o determinante:<\/p>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n2 \\cdot 7 \\cdot 1 + 1 \\cdot 1 \\cdot x + 1 \\cdot 4 \\cdot y &#8211; (1 \\cdot 7 \\cdot x + 2 \\cdot 1 \\cdot y + 1 \\cdot 4 \\cdot 1) = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n14 + x + 4y &#8211; 7x &#8211; 2y &#8211; 4 = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n-6x + 2y + 10 = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<p>Dividindo toda a equa\u00e7\u00e3o por 2:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + y + 5 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"exemplo-2\">Exemplo 2<\/h4>\n\n\n\n<p>Analisemos a equa\u00e7\u00e3o geral da reta 4x + y &#8211; 5 = 0 e verifiquemos quais pontos pertencem \u00e0 reta:<\/p>\n\n\n\n<p>Para o ponto <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$D(-1, 9)$<\/span><span class=\"latex-inline\">$<\/span>:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4(-1) + 9 &#8211; 5 = -4 + 9 &#8211; 5 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>Portanto, o ponto (<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$-1, 9$<\/span><span class=\"latex-inline\">$<\/span>) pertence \u00e0 reta.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"exemplos-pr\u00e1ticos-e-resolu\u00e7\u00e3o-de-exerc\u00edcios\"><span class=\"ez-toc-section\" id=\"Exemplos-praticos-e-resolucao-de-exercicios\"><\/span>Exemplos pr\u00e1ticos e resolu\u00e7\u00e3o de exerc\u00edcios<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>A melhor forma de consolidar o entendimento sobre a equa\u00e7\u00e3o geral da reta na Geometria Anal\u00edtica \u00e9 atrav\u00e9s da pr\u00e1tica. Para isso, resolveremos alguns exemplos para ilustrar o processo de encontrar a equa\u00e7\u00e3o de uma reta e verificar pontos pertencentes a ela.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exemplo-1-encontrando-a-equa\u00e7\u00e3o-da-reta\">Exemplo 1: Encontrando a equa\u00e7\u00e3o da reta<\/h3>\n\n\n\n<p>Quais s\u00e3o as coordenadas dos pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B$<\/span><span class=\"latex-inline\">$<\/span>?<br>Dado os pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A(2, 4)$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B(3, 7)$<\/span><span class=\"latex-inline\">$<\/span>, encontraremos a equa\u00e7\u00e3o da reta que passa por eles.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"passo-a-passo\">Passo a passo:<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Escrevemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\left| \\begin{array}{ccc}\n2 &#038; 3 &#038; x \\\n4 &#038; 7 &#038; y \\\n1 &#038; 1 &#038; 1 \\\n\\end{array} \\right| = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>Desenvolvemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n2 \\cdot 7 \\cdot 1 + 4 \\cdot 1 \\cdot x + 1 \\cdot 3 \\cdot y &#8211; 1 \\cdot 7 \\cdot x &#8211; 2 \\cdot 1 \\cdot y &#8211; 4 \\cdot 3 \\cdot 1 = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>Simplificamos:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[14 + 4x + 3y &#8211; 7x &#8211; 2y &#8211; 12 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li>Reduzimos os termos semelhantes:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + y + 2 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>Portanto, a equa\u00e7\u00e3o geral da reta \u00e9:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + y + 2 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exemplo-2-analisando-uma-reta-no-gr\u00e1fico\">Exemplo 2: Analisando uma reta no gr\u00e1fico<\/h3>\n\n\n\n<p>Vamos considerar uma reta representada no plano cartesiano e identificar a sua equa\u00e7\u00e3o. Suponhamos que a reta contenha os pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A(2, 1)$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B(5, 4)$<\/span><span class=\"latex-inline\">$<\/span>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"passo-a-passo-1\">Passo a passo:<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Escrevemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\left| \\begin{array}{ccc}\n2 &#038; 5 &#038; x \\\n1 &#038; 4 &#038; y \\\n1 &#038; 1 &#038; 1 \\\n\\end{array} \\right| = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>Desenvolvemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n2 \\cdot 4 \\cdot 1 + 1 \\cdot 1 \\cdot x + 1 \\cdot 5 \\cdot y &#8211; 1 \\cdot 4 \\cdot x &#8211; 2 \\cdot 1 \\cdot y &#8211; 1 \\cdot 5 \\cdot 1 = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>Simplificamos:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[8 + x + 5y &#8211; 4x &#8211; 2y &#8211; 5 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li>Reduzimos os termos semelhantes:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + 3y + 3 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>Podemos dividir todos os termos por 3:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-x + y + 1 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>A equa\u00e7\u00e3o geral da reta \u00e9:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-x + y + 1 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exerc\u00edcios-resolvidos\">Exerc\u00edcios resolvidos<\/h3>\n\n\n\n<p>Vamos resolver alguns exerc\u00edcios para fixar o conceito.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"quest\u00e3o-1\">Quest\u00e3o 1<\/h4>\n\n\n\n<p>Dado os pontos <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$A(2, 1)$<\/span><span class=\"latex-inline\">$<\/span> e <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$B(4, 7)$<\/span><span class=\"latex-inline\">$<\/span>, qual \u00e9 a equa\u00e7\u00e3o geral da reta?<\/p>\n\n\n\n<p><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Escrevemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n\\left| \\begin{array}{ccc}\n2 &#038; 4 &#038; x \\\n1 &#038; 7 &#038; y \\\n1 &#038; 1 &#038; 1 \\\n\\end{array} \\right| = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>Desenvolvemos o determinante:<\/li>\n<\/ol>\n\n\n\n<span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[\n2 \\cdot 7 \\cdot 1 + 1 \\cdot 1 \\cdot x + 1 \\cdot 4 \\cdot y &#8211; 1 \\cdot 7 \\cdot x &#8211; 2 \\cdot 1 \\cdot y &#8211; 1 \\cdot 4 \\cdot 1 = 0\n]$<\/span><span class=\"latex-inline\">$<\/span>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>Simplificamos:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[14 + x + 4y &#8211; 7x &#8211; 2y &#8211; 4 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li>Reduzimos os termos semelhantes:<\/li>\n<\/ol>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-6x + 2y + 10 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>Dividimos todos os termos por <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$2$<\/span><span class=\"latex-inline\">$<\/span>:<\/p>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[-3x + y + 5 = 0]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<p>A resposta correta \u00e9:<\/p>\n\n\n\n<p><strong>Alternativa D) <\/strong><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$-3x + y + 5 = 0$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"quest\u00e3o-2\">Quest\u00e3o 2<\/h4>\n\n\n\n<p>Verifique se os pontos abaixo pertencem \u00e0 equa\u00e7\u00e3o geral da reta <span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$4x + y &#8211; 5 = 0$<\/span><span class=\"latex-inline\">$<\/span>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A) (2, 0)<\/li>\n\n\n\n<li>B) (3, -3)<\/li>\n\n\n\n<li>C) (1, -1)<\/li>\n\n\n\n<li>D) (-1, 9)<\/li>\n\n\n\n<li>E) (0, -5)<\/li>\n<\/ul>\n\n\n\n<p><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p>Substitu\u00edmos os valores para verificar se a equa\u00e7\u00e3o \u00e9 satisfeita.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Para (2, 0):<\/li>\n<\/ul>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4 \\cdot 2 + 0 &#8211; 5 = 4 \\cdot 2 &#8211; 5 = 8 &#8211; 5 = 3 \\quad (\\text{falsa})]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Para (3, -3):<\/li>\n<\/ul>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4 \\cdot 3 + (-3) &#8211; 5 = 12 &#8211; 3 &#8211; 5 = 4 \\quad (\\text{falsa})]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Para (1, -1):<\/li>\n<\/ul>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4 \\cdot 1 + (-1) &#8211; 5 = 4 &#8211; 1 &#8211; 5 = -2 \\quad (\\text{falsa})]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Para (-1, 9):<\/li>\n<\/ul>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4 \\cdot (-1) + 9 &#8211; 5 = -4 + 9 &#8211; 5 = 0 \\quad (\\text{verdadeira})]$<\/span><span class=\"latex-inline\">$<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Para (0, -5):<\/li>\n<\/ul>\n\n\n\n<p><span class=\"latex-block\">$<\/span><span class=\"latex-inline\">$[4 \\cdot 0 + (-5) &#8211; 5 = -5 &#8211; 5 = -10 \\quad (\\text{falsa})]$<\/span>$<\/span><\/p>\n\n\n\n<p>Portanto:<\/p>\n\n\n\n<p><strong>Alternativa correta: D) (-1, 9)<\/strong><\/p>\n\n\n\n<p>Esses exemplos pr\u00e1ticos ilustram como utilizar a equa\u00e7\u00e3o geral da reta para resolver problemas espec\u00edficos na Geometria Anal\u00edtica. Praticar esses exerc\u00edcios \u00e9 fundamental para dominar a aplica\u00e7\u00e3o e a compreens\u00e3o desse conceito matem\u00e1tico.<\/p>\n\n\n\n<p>Agora que voc\u00ea j\u00e1 sabe tudo sobre a equa\u00e7\u00e3o geral da reta, prepare-se para os concursos militares com o l\u00edder no que vale: classifica\u00e7\u00e3o! Clique no banner e adquira nossos cursos!<\/p>\n\n\n\n<a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/militares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/militares.estrategia.com\/portal\/wp-content\/uploads\/2022\/08\/EM-cta-geral.jpeg\" alt=\"CTA - Militares\" title=\"EM-cta-geral\">\n        <\/a>\n\n\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\">Progress\u00e3o Aritm\u00e9tica (PA): o que \u00e9, tipos e propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/\">Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\">Par ordenado e produto cartesiano: o que s\u00e3o e suas propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/\">O que \u00e9 uma Rela\u00e7\u00e3o Bin\u00e1ria na Matem\u00e1tica<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"A equa\u00e7\u00e3o geral da reta, ax + by + c = 0, \u00e9 fundamental na geometria anal\u00edtica para descrever retas no plano cartesiano atrav\u00e9s das coordenadas x e y.\n","protected":false},"author":11,"featured_media":79067,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[307],"tags":[64,313,162,161,167,271],"wl_entity_type":[187],"class_list":{"0":"post-77643","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"tag-marinhadobrasil","9":"tag-313","10":"tag-carreiramilitar","11":"tag-concursosmilitares","12":"tag-exercito","13":"tag-fab","14":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Equa\u00e7\u00e3o geral da reta na Geometria Anal\u00edtica<\/title>\n<meta name=\"description\" content=\"A equa\u00e7\u00e3o geral da reta \u00e9 fundamental na geometria anal\u00edtica para descrever retas no plano cartesiano 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