{"id":28869,"date":"2021-04-20T13:00:28","date_gmt":"2021-04-20T13:00:28","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28869"},"modified":"2024-09-22T02:07:52","modified_gmt":"2024-09-22T02:07:52","slug":"equacao-da-reta-e-os-sistemas-cartesianos","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/equacao-da-reta-e-os-sistemas-cartesianos\/","title":{"rendered":"Equa\u00e7\u00e3o da Reta e os Sistemas Cartesianos"},"content":{"rendered":"<p><center><\/p>\n<h1>Equa\u00e7\u00e3o da Reta e os Sistemas Cartesianos<\/h1>\n<p style=\"text-align: center;\"><em><strong>Resumo:<\/strong><br \/>\n      Nesta aula, abordaremos os fundamentos da geometria anal\u00edtica, mostrando como representar pontos em um plano usando coordenadas e como formular a equa\u00e7\u00e3o da reta a partir da inclina\u00e7\u00e3o e de um ponto dado. Conceitos chave como a inclina\u00e7\u00e3o, o uso da equa\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span> e a representa\u00e7\u00e3o gr\u00e1fica de retas s\u00e3o explorados, junto com exerc\u00edcios pr\u00e1ticos e aplica\u00e7\u00f5es para resolver problemas do mundo real, como o c\u00e1lculo de posi\u00e7\u00f5es e interse\u00e7\u00f5es de retas.<\/em><\/p>\n<p>   <strong>Objetivos de Aprendizagem<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Compreender<\/strong> os princ\u00edpios b\u00e1sicos da geometria anal\u00edtica e sua aplica\u00e7\u00e3o na representa\u00e7\u00e3o de pontos em um plano cartesiano.<\/li>\n<li><strong>Identificar<\/strong> a f\u00f3rmula da inclina\u00e7\u00e3o de uma reta e seu significado geom\u00e9trico.<\/li>\n<li><strong>Aplicar<\/strong> a equa\u00e7\u00e3o geral da reta <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span> para descrever rela\u00e7\u00f5es lineares.<\/li>\n<li><strong>Calcular<\/strong> a equa\u00e7\u00e3o da reta a partir de um ponto e da inclina\u00e7\u00e3o.<\/li>\n<li><strong>Desenhar<\/strong> retas em um plano cartesiano usando sua equa\u00e7\u00e3o linear.<\/li>\n<li><strong>Resolver<\/strong> problemas envolvendo a interse\u00e7\u00e3o de duas retas usando sistemas de equa\u00e7\u00f5es.<\/li>\n<li><strong>Analisar<\/strong> a rela\u00e7\u00e3o entre duas magnitudes lineares e como represent\u00e1-las por meio de uma equa\u00e7\u00e3o de reta.<\/li>\n<\/ol>\n<p>   <strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n   <a href=\"#1\">Os princ\u00edpios da Geometria Anal\u00edtica<\/a><br \/>\n   <a href=\"#2\">A Equa\u00e7\u00e3o da Reta<\/a><br \/>\n   <a href=\"#3\">Como desenhar a Equa\u00e7\u00e3o da Reta<\/a><br \/>\n   <a href=\"#4\">Interse\u00e7\u00f5es entre Retas<\/a>\n   <\/p>\n<p><\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mNISGHOByAI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">Agora come\u00e7aremos nosso estudo sobre a equa\u00e7\u00e3o da reta, os sistemas cartesianos e os princ\u00edpios da geometria anal\u00edtica.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Os princ\u00edpios da Geometria Anal\u00edtica<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><strong>Quando os n\u00fameros reais s\u00e3o introduzidos<\/strong><\/a>, normalmente se diz que esses s\u00e3o pontos sobre uma reta<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"RETA DOS REAIS\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"RETA DOS REAIS\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">A partir disso, Descartes teve a genialidade de usar duas retas para representar pontos em um plano como um par de coordenadas (x, y)<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><\/noscript><\/p>\n<p><a name=\"2\"><\/a>     <\/p>\n<h2>A Equa\u00e7\u00e3o da Reta<\/h2>\n<p style=\"text-align: justify;\">Usando esses conceitos, agora \u00e9 poss\u00edvel considerar um conjunto de pontos no plano para formar curvas no plano, onde \u00e0 coordenada <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> corresponde outra coordenada <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, e essa regra de correspond\u00eancia \u00e9 dada por uma fun\u00e7\u00e3o. \u00c9 neste ponto que a \u00e1lgebra penetra na geometria e nasce a \u00abGeometria Anal\u00edtica\u00bb.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=320s\" target=\"_blank\" rel=\"noopener\"><strong>Geometricamente, entendemos uma reta<\/strong><\/a> como a curva que conecta dois pontos percorrendo a dist\u00e2ncia mais curta poss\u00edvel.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"RETA NO PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"RETA NO PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Geometricamente, entendemos uma reta como a curva que conecta dois pontos percorrendo a dist\u00e2ncia mais curta poss\u00edvel. E analisando isso, em virtude do teorema de Tales, veremos que a cada incremento da coordenada <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> corresponde um incremento da coordenada <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, tal que o quociente <span class=\"katex-eq\" data-katex-display=\"false\">m=(y_2 - y_1)\/(x_2 - x_1)=\\Delta y \/ \\Delta x<\/span> ser\u00e1 sempre constante para qualquer par de pontos sobre a reta. Isso \u00e9 o que chamamos de <strong>\u00abinclina\u00e7\u00e3o da reta\u00bb.<\/strong><\/p>\n<p style=\"text-align: justify;\">Como a inclina\u00e7\u00e3o \u00e9 a mesma para qualquer par de pontos da reta, ent\u00e3o se considerarmos os pontos da reta com coordenadas <span class=\"katex-eq\" data-katex-display=\"false\">(x, y),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">(x_2, y_2),<\/span> podemos escrever:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{y - y_0}{x - x_0} = \\frac{y_2 - y_1}{x_2 - x_1}<\/span>\n<p style=\"text-align: justify;\">O que \u00e9 o mesmo que dizer<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{matrix}y &amp; = &amp; \\displaystyle \\frac{y_2 - y_1}{x_2 - x_1} (x - x_0 ) + y_0 \\\\ \\\\ &amp; = &amp; \\displaystyle \\frac{\\Delta y}{\\Delta x} (x - x_0) + y_0 \\end{matrix}<\/span>\n<p style=\"text-align: justify;\">\u00c9 daqui que vem a conhecida <strong>equa\u00e7\u00e3o da reta<\/strong><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{red}{{y = m(x - x_0) + y_0}}<\/span>\n<p style=\"text-align: justify;\">Aqui, o par <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0)<\/span> \u00e9 um ponto fixo, enquanto o par <span class=\"katex-eq\" data-katex-display=\"false\">(x, y)<\/span> \u00e9 um ponto qualquer.<\/p>\n<h3>Exerc\u00edcios de exemplo<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Calcular a equa\u00e7\u00e3o da reta que passa pelo ponto <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = (2, 3)<\/span> com inclina\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">m = 3\/2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=695s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<li>Calcular a equa\u00e7\u00e3o da reta que passa pelo ponto <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = (1, 8)<\/span> com inclina\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">m = 7\/5<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=750s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<li>Calcular a equa\u00e7\u00e3o da reta que passa pelos pontos <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1) = (3, 5)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">(x_2, y_2) = (1, -2)<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=818s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Como desenhar a Equa\u00e7\u00e3o da Reta<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1063s\" target=\"_blank\" rel=\"noopener\"><strong>J\u00e1 vimos como obter<\/strong><\/a> a equa\u00e7\u00e3o da reta a partir de alguma informa\u00e7\u00e3o gr\u00e1fica; agora seguiremos o caminho inverso, obtendo a representa\u00e7\u00e3o gr\u00e1fica a partir da equa\u00e7\u00e3o da reta.<\/p>\n<p style=\"text-align: justify;\">No fim das contas, a equa\u00e7\u00e3o da reta sempre termina sendo apresentada da seguinte forma.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span>\n<p style=\"text-align: justify;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">m = \\Delta Y \/ \\Delta x<\/span> \u00e9 a inclina\u00e7\u00e3o e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u00e9 o coeficiente de posi\u00e7\u00e3o. A partir disso, temos a seguinte figura<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"RETA NO PLANO CARTESIANO COM COORDENADAS\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"RETA NO PLANO CARTESIANO COM COORDENADAS\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><\/noscript><\/p>\n<h3>Exerc\u00edcio de Exemplo<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Desenhar a reta da equa\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle \\frac{3}{4}x + 2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1139s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<li>Desenhar a reta da equa\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle -\\frac{2}{5}x + 6<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1209s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<\/ol>\n<h3>Problemas de aplica\u00e7\u00e3o da equa\u00e7\u00e3o da reta<\/h3>\n<p style=\"text-align: justify;\">A reta pode ser usada para resolver problemas que envolvem a rela\u00e7\u00e3o direta entre duas magnitudes, como \u00e9 o caso dos seguintes exemplos<\/p>\n<ol style=\"text-align: justify;\">\n<li>Um ve\u00edculo com posi\u00e7\u00e3o inicial <span class=\"katex-eq\" data-katex-display=\"false\">x_0 = 12[m]<\/span> move-se com uma velocidade de <span class=\"katex-eq\" data-katex-display=\"false\">v=0,3[m\/s]<\/span>. Qual ser\u00e1 sua posi\u00e7\u00e3o ap\u00f3s <span class=\"katex-eq\" data-katex-display=\"false\">30[s]<\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1257s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<li>Uma pessoa vai \u00e0 feira e compra <span class=\"katex-eq\" data-katex-display=\"false\">1[kg]<\/span> de ma\u00e7\u00e3s, gastando um total de <span class=\"katex-eq\" data-katex-display=\"false\">50 Z\\$.<\/span> Nesse mesmo dia, a pessoa voltou \u00e0 feira para comprar outros <span class=\"katex-eq\" data-katex-display=\"false\">3[kg]<\/span> de ma\u00e7\u00e3s, gastando um total de <span class=\"katex-eq\" data-katex-display=\"false\">60 Z\\$.<\/span>. Qual \u00e9 o pre\u00e7o das ma\u00e7\u00e3s e o pre\u00e7o das passagens?<a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1383s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Interse\u00e7\u00f5es entre Retas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1855s\" target=\"_blank\" rel=\"noopener\"><strong>Suponhamos que temos duas retas<\/strong><\/a> e queremos encontrar o ponto comum entre elas; isto \u00e9, encontrar a interse\u00e7\u00e3o entre retas. Para resolver este tipo de problema, devemos resolver um sistema de equa\u00e7\u00f5es. Para entender isso melhor, vejamos o seguinte exemplo.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1914s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremos as seguintes retas:<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_1 \\; : \\; y= \\displaystyle \\frac{3}{2}x + 1<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_1 \\; : \\; y=\\displaystyle -\\frac{1}{3}x + 9<\/span>\n<p style=\"text-align: justify;\">Onde essas duas retas se intersectam?<\/p>\n<p style=\"text-align: justify;\">Para resolver isso, fazemos o seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle \\frac{3}{2}x + 1<\/span><\/td>\n<td>; Reta <span class=\"katex-eq\" data-katex-display=\"false\">L_1<\/span><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; Reta <span class=\"katex-eq\" data-katex-display=\"false\">L_2<\/span><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x + 1 = -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; De (1) e (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x = -\\frac{1}{3}x + 8<\/span><\/td>\n<td>; Subtraindo 1 de ambos os lados<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">9x = -2x + 48<\/span><\/td>\n<td>; Multiplicando ambos os lados por 6<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">11x =48<\/span><\/td>\n<td>; Somando 2x a ambos os lados<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x = \\frac{48}{11}<\/span><\/td>\n<td>; Dividindo ambos os lados por 11<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{2}\\cdot \\frac{48}{11} + 1<\/span><\/td>\n<td>; De (1) e (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{1}\\cdot \\frac{24}{11} + \\frac{11}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle \\frac{83}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x, y)= \\left(\\frac{48}{11}, \\frac{83}{11} \\right)<\/span><\/td>\n<td>; De (3) e (4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Portanto, o ponto de interse\u00e7\u00e3o entre as retas \u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">(x, y)= \\displaystyle \\left(\\frac{48}{11}, \\frac{83}{11} \\right).<\/span>\n<h3>Exemplo de problemas de aplica\u00e7\u00e3o para interse\u00e7\u00e3o entre retas<\/h3>\n<p style=\"text-align: justify;\">Para uma festa, foram vendidos 600 ingressos com uma arrecada\u00e7\u00e3o total de <span class=\"katex-eq\" data-katex-display=\"false\">\\$1.300.000.<\/span> Os ingressos para jovens foram vendidos a <span class=\"katex-eq\" data-katex-display=\"false\">\\$1.000,<\/span> e os ingressos para adultos a <span class=\"katex-eq\" data-katex-display=\"false\">\\$3.000.<\/span> Quantos adultos e jovens foram \u00e0 festa?<a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=2255s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLU\u00c7\u00c3O]<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equa\u00e7\u00e3o da Reta e os Sistemas Cartesianos Resumo: Nesta aula, abordaremos os fundamentos da geometria anal\u00edtica, mostrando como representar pontos em um plano usando coordenadas e como formular a equa\u00e7\u00e3o da reta a partir da inclina\u00e7\u00e3o e de um ponto dado. Conceitos chave como a inclina\u00e7\u00e3o, o uso da equa\u00e7\u00e3o e a representa\u00e7\u00e3o gr\u00e1fica de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28865,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":20,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-28869","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-e-geometria","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Equa\u00e7\u00e3o da Reta e os Sistemas Cartesianos - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda os fundamentos da Equa\u00e7\u00e3o da Reta e dos Sistemas Cartesianos. 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