{"id":66322,"date":"2022-04-19T08:16:22","date_gmt":"2022-04-19T11:16:22","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=66322"},"modified":"2024-03-08T17:17:55","modified_gmt":"2024-03-08T20:17:55","slug":"sistemas-lineares","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/","title":{"rendered":"Sistemas lineares: o que s\u00e3o, como resolver e aplica\u00e7\u00f5es"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Imagine que voc&ecirc; possui duas <span id=\"urn:enhancement-2293cd7b-5cc3-4a46-b8b9-390e32231013\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>, com duas inc&oacute;gnitas e a sua miss&atilde;o &eacute; descobrir o valor que torna ambas verdadeiras. Essa &eacute; a ideia que norteia todo o estudo dos sistemas lineares, um tipo de ferramenta <span id=\"urn:enhancement-3102a748-e775-4819-9484-89556d124a37\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> muito frequente nas provas de vestibular.<p class=\"wp-block-paragraph\">Veja, a seguir, como s&atilde;o definidos os sistemas lineares e como podem ser classificados. Acompanhe tamb&eacute;m diferentes maneiras de resolver e encontrar o resultado das <span id=\"urn:enhancement-18fb403b-5f91-43ae-a3b3-5d0f749f7c8c\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>. Vamos l&aacute;!&nbsp;<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 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<\/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:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#O-que-sao-sistemas-lineares\" >O que s&atilde;o sistemas lineares?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#Classificacao-dos-sistemas-lineares\" >Classifica&ccedil;&atilde;o dos sistemas lineares<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#Representacao-matricial-de-um-sistema-linear\" >Representa&ccedil;&atilde;o matricial de um sistema linear<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#Como-resolver-sistemas-lineares\" >Como resolver sistemas lineares?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#Estude-para-o-Enem-com-o-Estrategia-Vestibulares\" >Estude para o Enem com o Estrat&eacute;gia Vestibulares!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-sistemas-lineares\"><span class=\"ez-toc-section\" id=\"O-que-sao-sistemas-lineares\"><\/span>O que s&atilde;o sistemas lineares?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os sistemas lineares podem ser definidos como um conjunto de N <span id=\"urn:enhancement-26eb53fd-b07a-4303-a01c-5e90c105e98c\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> que possuem, juntas, N inc&oacute;gnitas. Por exemplo, um sistema com 3 <span id=\"urn:enhancement-a77b0fb5-9970-4dfa-b810-848bced2a769\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> e 3 inc&oacute;gnitas, como o mostrado a seguir. Para sinalizar que as <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/equacoes\/\">equa&ccedil;&otilde;es <\/a>fazem parte de um sistema linear, &eacute; necess&aacute;rio adicionar o s&iacute;mbolo matem&aacute;tico da chave.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/1jUbxc4TdWsLV4bHOlTFdD7z6Jo_rbWiS4OZfAyeKmXioDvbVPhmRfT68iseOotDKMCIpywF9kMMqiC53sHmYZTTVWjKdoPLuSRdwVZEvvzpSETTE07xIcpBBMKQZja91MOP9D0g\" alt=\"sistemas lineares - exemplo\"><\/figure>\n<\/div><h3 class=\"wp-block-heading\" id=\"h-equacoes-lineares\">Equa&ccedil;&otilde;es lineares<\/h3><p class=\"wp-block-paragraph\">Para compreender melhor os sistemas, &eacute; importante conhecer as <span id=\"urn:enhancement-7af3268d-4249-43d8-82b4-3e5e17f138b7\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> lineares. Elas s&atilde;o <span id=\"urn:enhancement-71949b67-36b5-4b04-accf-c91b458432f6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> <span id=\"urn:enhancement-fc5f81a8-b6e6-40c2-9132-1c6869d4f347\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> constitu&iacute;das de diferentes termos. Eles s&atilde;o definidos pela <span id=\"urn:enhancement-d7d2cdd9-cf8c-4e39-853f-b45e321365f5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> entre um <span id=\"urn:enhancement-ceb8d198-7113-4565-b975-fb59a238218f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span> a<sub>m<\/sub> e uma inc&oacute;gnita qualquer.&nbsp;<\/p><p class=\"wp-block-paragraph\">Os termos s&atilde;o somados entre si para constituir a <span id=\"urn:enhancement-6d620e31-9069-4ef6-a574-1723b8843db5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>. No exemplo acima, temos 1x, 3y, 1z e assim por diante. Quando o termo n&atilde;o apresenta inc&oacute;gnita, mas somente um n&uacute;mero &ldquo;isolado&rdquo;, consideramos que &eacute; um termo independente.<\/p><p class=\"wp-block-paragraph\">A representa&ccedil;&atilde;o de uma <span id=\"urn:enhancement-3d6429b3-3174-4d9d-abe0-0310f84b5bd2\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> linear pode ser dada por:&nbsp;<\/p><p class=\"wp-block-paragraph\">a<sub>1<\/sub>.x<sub>1<\/sub> + a<sub>2<\/sub>.x<sub>2<\/sub> + a<sub>3<\/sub>.x<sub>3<\/sub> + a<sub>4<\/sub>.x<sub>4<\/sub> + &hellip; + a<sub>n<\/sub>.x<sub>n<\/sub> = b<sub>n<\/sub><\/p><p class=\"wp-block-paragraph\">Ao considerar as defini&ccedil;&otilde;es acima, pode-se concluir que a melhor <span id=\"urn:enhancement-d4d74069-557f-485f-8d74-84eff3edd604\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> gen&eacute;rica de representar um sistema linear &eacute; a que est&aacute; na imagem abaixo. Na qual s&atilde;o descritos todos os <span id=\"urn:enhancement-1a266260-b0ad-41ab-a1a8-07a7f0fd0003\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> (a<sub>nn<\/sub>), as diferentes inc&oacute;gnitas e o termo independente (b).<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/NAGHP5hCU35SXd6U5cjJuAYPPPB2ERKt-hCafaWrRs5Uct2gdXXi8hoKSjEaNzQKUyJovrc0hwRrqWoci6qrlehf0KNk1YqQ4UTDeqSs0jghsPgcx54CpEva0lfatYktozLKgeU6\" alt=\"\"><\/figure>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-classificacao-dos-sistemas-lineares\"><span class=\"ez-toc-section\" id=\"Classificacao-dos-sistemas-lineares\"><\/span>Classifica&ccedil;&atilde;o dos sistemas lineares<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os sistemas lineares podem ser classificados de diferentes formas. Abaixo vamos entender quando &eacute; considerado um sistema determinado, indeterminado ou imposs&iacute;vel.<\/p><h3 class=\"wp-block-heading\" id=\"h-sistema-possivel-e-determinado-spd\">Sistema poss&iacute;vel e determinado (SPD)<\/h3><p class=\"wp-block-paragraph\">Um sistema linear determinado &eacute; aquele em que as inc&oacute;gnitas possuem apenas um valor real que as satisfazem. O resultado &eacute; obtido com uma solu&ccedil;&atilde;o ou um conjunto solu&ccedil;&atilde;o que est&aacute; contido entre os n&uacute;meros reais.&nbsp;<\/p><p class=\"wp-block-paragraph\">O fato de considerar um sistema poss&iacute;vel, indica que ele pode ser resolvido e possui uma resposta concreta. Veja um exemplo:<\/p><p class=\"wp-block-paragraph\">Para resolver esse sistema de <span id=\"urn:enhancement-91ca5f2a-ba2c-4913-8fe9-86d483e4f38c\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>, podemos isolar uma das inc&oacute;gnitas em uma das <span id=\"urn:enhancement-2c526e56-cd10-408f-83ec-d6ba64e58f51\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span>.<\/p><p class=\"wp-block-paragraph\">vamos partir do isolamento de x, na <span id=\"urn:enhancement-47b6b161-8d19-43d5-9273-8026cb34d120\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> que est&aacute; na parte debaixo da chave, assim:<\/p><p class=\"wp-block-paragraph\">z &ndash; x = 2<br>&ndash; x = 2 &ndash; z<br>x = -2 +z&nbsp;<br><strong>x = z &ndash; 2<\/strong><\/p><p class=\"wp-block-paragraph\">Com x isolado, podemos substituir seu valor na primeira <span id=\"urn:enhancement-2c756819-b70c-46a8-8648-ad78003c6828\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> linear da chave:<\/p><p class=\"wp-block-paragraph\">x + 2z = 19<br>( z &ndash; 2) + 2z = 19<br>z &ndash; 2 + 2z = 19&nbsp;<br>z + 2z &ndash; 2 = 19&nbsp;<br>z + 2z = 19 + 2<br>3z = 21<br><strong>z = 7<\/strong><\/p><p class=\"wp-block-paragraph\">Agora que encontramos o valor de z, voc&ecirc; j&aacute; pode substitu&iacute;-lo em qualquer uma das <span id=\"urn:enhancement-b9ac1f34-8e87-4554-93cf-af4445a7bda6\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> e o valor de x ser&aacute; naturalmente expresso:<\/p><p class=\"wp-block-paragraph\">x = z &ndash; 2<br>x = 7 &ndash; 2<br><strong>x = 5<\/strong><\/p><p class=\"wp-block-paragraph\">Perceba que cada uma das inc&oacute;gnitas admitiu um s&oacute; valor e existe apenas um conjunto real de n&uacute;meros que representam a resolu&ccedil;&atilde;o desse sistema:<strong> S = {5,7}<\/strong>.<\/p><h3 class=\"wp-block-heading\" id=\"h-sistema-possivel-e-indeterminado-spi\">Sistema poss&iacute;vel e indeterminado (SPI)<\/h3><p class=\"wp-block-paragraph\">No caso de sistemas em que mais de um elemento pode ser a resposta para uma s&oacute; inc&oacute;gnita, teremos um sistema poss&iacute;vel e indeterminado. Geralmente, admite-se que eles possuem infinitas solu&ccedil;&otilde;es.<\/p><p class=\"wp-block-paragraph\">&Eacute; o caso de sistemas lineares em que as <span id=\"urn:enhancement-730baeb4-dceb-4c95-81fa-0fd27cfff30e\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> lineares s&atilde;o congruentes entre si. Ou seja, apresentam todos os <span id=\"urn:enhancement-f8c69601-0785-423e-b381-611b3e39bb4e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> proporcionais, junto a suas inc&oacute;gnitas. Assim:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/W83rcnmoKNPVovh0Ltva_2fouubOUkGB30D6TIS_eqkb4HBP4NAz8elvRmDHMMxCgRMrrG6ma_LIFo0eDqIej06zAk_gvPTQeYQMPm_KBhun4p0cD-limw7s2UJKpKee5uF0uLpu\" alt=\"sistemas lineares determinados e poss&iacute;veis\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Observe que, se multiplicarmos a primeira <span id=\"urn:enhancement-c1bcd5d6-b3b9-4a11-a14c-ad730774d225\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> por dois, teremos exatamente a <span id=\"urn:enhancement-706419e6-5424-4a42-be51-761a0d14efe6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> demonstrada na linha de baixo. Nesse caso, fica imposs&iacute;vel determinar os valores de x e y, entretanto s&atilde;o poss&iacute;veis infinitas solu&ccedil;&otilde;es que tornem esse sistema linear real.<\/p><h3 class=\"wp-block-heading\" id=\"h-sistema-impossivel-si\">Sistema imposs&iacute;vel (SI)<\/h3><p class=\"wp-block-paragraph\">No caso dos sistemas imposs&iacute;veis, n&atilde;o existem solu&ccedil;&otilde;es reais que possam satisfazer as <span id=\"urn:enhancement-abe1fae0-2c0c-4a26-b864-400bdf134766\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>. Ou seja, os resultados seriam incongruentes entre si, no decorrer do c&aacute;lculo.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rOhpIczL8wd_lqDqqFCNm31r8DMeKhYmk3Rh_U_NwceDXtBjU_K9IVzh2cpIG8NAb2WwgnNINNmk77cVnq5fsozXh5yPI8zebV45gAKAAdNATE3NpnyuE9D6mD160oNgPCoeOnQt\" alt=\"sistemas lineares imposs&iacute;veis\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Veja como as <span id=\"urn:enhancement-0fb54f23-ae07-4d97-974f-bc748b005cc8\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> n&atilde;o podem ser resolvidas: a mesma <span id=\"urn:enhancement-490e1b01-55cc-4a66-8a94-ff5c6b2e3c47\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> resultando em valores diferentes &eacute; uma inconsist&ecirc;ncia <span id=\"urn:enhancement-8be33d20-76dd-4579-a613-8acef259870d\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> que torna o sistema imposs&iacute;vel. Ou seja, a solu&ccedil;&atilde;o dele &eacute; nula ou vazia: <strong>S = &Oslash;<\/strong>.<\/p><div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button has-custom-font-size is-style-fill\" style=\"font-size:30px\"><a class=\"wp-block-button__link has-luminous-vivid-amber-to-luminous-vivid-orange-gradient-background has-background\" href=\"https:\/\/vestibulares.estrategia.com\/catalogo-produtos\/\" target=\"_blank\" rel=\"noopener noreferrer\"><meta charset=\"utf-8\"><strong>Conhe&ccedil;a os cursos do Estrat&eacute;gia!<\/strong><\/a><\/div>\n<\/div><h2 class=\"wp-block-heading\" id=\"h-representacao-matricial-de-um-sistema-linear\"><span class=\"ez-toc-section\" id=\"Representacao-matricial-de-um-sistema-linear\"><\/span>Representa&ccedil;&atilde;o matricial de um sistema linear<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">&Eacute; poss&iacute;vel representar os sistemas lineares a partir de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/matrizes\/\" target=\"_blank\">matrizes<\/a>, que s&atilde;o ferramentas <span id=\"urn:enhancement-a91b0a22-9119-4338-be45-2b9337f1db3c\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;ticas<\/span> que imitam uma tabela, com elementos marcados com a nota&ccedil;&atilde;o a<sub>ij<\/sub>.&nbsp;<\/p><p class=\"wp-block-paragraph\">O a &eacute; o elemento propriamente dito, i &eacute; a linha da matriz em que ele se encontra e, por fim, j &eacute; a coluna de localiza&ccedil;&atilde;o do elemento. Um elemento a<sub>32<\/sub> est&aacute; na linha 3 e na coluna 2 de uma matriz A qualquer.&nbsp;<\/p><p class=\"wp-block-paragraph\">Para transferir a nota&ccedil;&atilde;o de matrizes para os sistemas lineares. Uma das tabelas deve representar os <span id=\"urn:enhancement-9bbb5729-4736-49d4-99d5-c18b2cac7260\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> das inc&oacute;gnitas, em outra teremos os <span id=\"urn:enhancement-b6f34233-3b82-4d00-85b1-1a52988945ab\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> distribu&iacute;dos em uma &uacute;nica coluna e, elas seriam multiplicadas. Como resultado dessa <span id=\"urn:enhancement-98edb1c3-5a3d-4f0e-99f7-2ad9af4a3db9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span>, teremos uma tabela de coluna &uacute;nica com todos os termos independentes. Assim:<\/p><h2 class=\"wp-block-heading\" id=\"h-como-resolver-sistemas-lineares\"><span class=\"ez-toc-section\" id=\"Como-resolver-sistemas-lineares\"><\/span>Como resolver sistemas lineares?<span class=\"ez-toc-section-end\"><\/span><\/h2><h3 class=\"wp-block-heading\" id=\"h-metodo-da-substituicao\">M&eacute;todo da substitui&ccedil;&atilde;o<\/h3><p class=\"wp-block-paragraph\">A resolu&ccedil;&atilde;o de sistemas lineares com o m&eacute;todo da substitui&ccedil;&atilde;o &eacute; aquela em que, isola-se a inc&oacute;gnita em uma das <span id=\"urn:enhancement-72fcbd17-d28b-45d0-8b6a-8992c9d2119a\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> e depois a inc&oacute;gnita &eacute; substitu&iacute;da pela <span id=\"urn:enhancement-a513805e-a12d-4be7-a736-50291b03146d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o alg&eacute;brica<\/span> que a representa nas outras <span id=\"urn:enhancement-cd921422-0214-4726-98f6-d9913c86134a\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> do sistema.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/2cGpK0uUFzaAyE2LYH9uxsr3UTKXODDgwnIDHeoFWU5G2Y6CJPXxzrv8LhiC-R_xp1a4hCBiA8zabA_-IVHkBZZKMfJ3AwUambXk0081sR5tNjEHz2tSi6VVZC4M6aT9qP43MdaN\" alt=\"sistemas lineares m&eacute;todos de substitui&ccedil;&atilde;o\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Para escolher qual inc&oacute;gnita iremos isolar, pode-se escolher aquela que aparece com <span id=\"urn:enhancement-cf100c3d-6893-4377-9a19-b3da1d97c838\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span> 1 em v&aacute;rias das <span id=\"urn:enhancement-23de1a49-d956-4615-8e59-d35c95a4f8f3\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> lineares. Assim, optamos por partir do isolamento da inc&oacute;gnita x:<\/p><p class=\"wp-block-paragraph\">x + 5z = 52<br><strong>x = 52 &ndash; 5z<\/strong><\/p><p class=\"wp-block-paragraph\">Com isso em m&atilde;os, podemos manipular a primeira <span id=\"urn:enhancement-c8d33672-dd46-4509-8499-0752606a190f\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, substituindo o valor de x por 52 &ndash; 5z:<\/p><p class=\"wp-block-paragraph\">x + y + 2z = 33<br>(52 &ndash; 5z) + y + 2z = 33<br>52 &ndash; 5z + y + 2z = 33<\/p><p class=\"wp-block-paragraph\">Agora, vamos agrupar todas as inc&oacute;gnitas iguais em um s&oacute; termo. Bem como faremos as opera&ccedil;&otilde;es necess&aacute;rias com o termo independente, de <span id=\"urn:enhancement-474ca249-7594-4fbb-bce1-abae7c88ec0a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> que alcan&ccedil;aremos um c&aacute;lculo mais curto:<\/p><p class=\"wp-block-paragraph\">52 &ndash; 5z + y + 2z = 33&nbsp;<br>&ndash; 5z + 2z + y = 33 &ndash; 52<br>&ndash; 3z + y = &ndash; 19<\/p><p class=\"wp-block-paragraph\">Como &eacute; mais intuitivo trabalhar com valores positivos, podemos multiplicar toda a <span id=\"urn:enhancement-cb4f849c-4f02-425b-9b02-f49193ab18c7\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> por -1:<\/p><p class=\"wp-block-paragraph\">3z &ndash; y = 19&nbsp;<\/p><p class=\"wp-block-paragraph\">Partimos, ent&atilde;o, para o isolamento da inc&oacute;gnita y:<\/p><p class=\"wp-block-paragraph\"><strong>3z &ndash; 19 = y<\/strong><\/p><p class=\"wp-block-paragraph\">Na segunda <span id=\"urn:enhancement-7308dc95-ae8a-4165-a447-1d63289a93b5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> do sistema linear, atribu&iacute;mos o valor 3z &ndash; 19 ao termo y:<\/p><p class=\"wp-block-paragraph\">&nbsp;y + z = 17&nbsp;<br>3z &ndash; 19 + z = 17<br>3z + z = 17 + 19<br>4z = 36&nbsp;<br><strong>z = 9&nbsp;<\/strong><\/p><p class=\"wp-block-paragraph\">Agora que encontramos o valor de z, basta substitu&iacute;-lo na <span id=\"urn:enhancement-ecc1cb7a-f154-47cb-bdf5-ff7b4a2d97d1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> de isolamento de y:<\/p><p class=\"wp-block-paragraph\">3z &ndash; 19 = y<br>3.(9) &ndash; 19 = y&nbsp;<br>27 &ndash; 19 = y<br><strong>y=8<\/strong><\/p><p class=\"wp-block-paragraph\">Por fim, substitui-de o valor de z na <span id=\"urn:enhancement-49226546-7787-46d0-94be-7d3ed40e1676\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> que relaciona o valor de x e de z:<\/p><p class=\"wp-block-paragraph\">x = 52 &ndash; 5z<br>x = 52 &ndash; 5.(9)<br>x = 52 &ndash; 45<br><strong>x = 7<\/strong><\/p><p class=\"wp-block-paragraph\">Assim, apenas com a manipula&ccedil;&atilde;o e substitui&ccedil;&atilde;o das <span id=\"urn:enhancement-7acd7b1b-f94d-4e72-a7b7-294c955012b2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> encontramos os valores de x, y e z, que comp&otilde;em o conjunto solu&ccedil;&atilde;o: <strong>S = {7,8,9}. <\/strong>Esse &eacute; um sistema que n&atilde;o admite m&uacute;ltiplas solu&ccedil;&otilde;es e, por isso, &eacute; poss&iacute;vel e determinado (<strong>SPD<\/strong>).<\/p><h3 class=\"wp-block-heading\" id=\"h-metodo-da-soma-ou-da-adicao\">M&eacute;todo da soma ou da adi&ccedil;&atilde;o<\/h3><p class=\"wp-block-paragraph\">No m&eacute;todo da <span id=\"urn:enhancement-4a736810-461b-4b1b-aadc-e3dd6553d97a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span>, manipulam-se as <span id=\"urn:enhancement-29b26737-f358-49b3-89ff-ad0cfce3f7af\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>, de maneira que a <span id=\"urn:enhancement-e05eb973-064e-40ab-ab9c-3329526c5d2c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">adi&ccedil;&atilde;o<\/span> entre elas forne&ccedil;a inc&oacute;gnitas com <span id=\"urn:enhancement-857faa62-2165-4d0f-9321-2fce0306a14b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> nulos, ou seja, deixe uma inc&oacute;gnita isolada. Acompanhe o <span id=\"urn:enhancement-f54b351b-ed4a-4321-9be5-3a61abd36e73\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/desenvolvimento\">desenvolvimento<\/span>:<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/T0xU1nxud98jl8ea0QOT79RQRtBSAeFMxAd1LNg_sTkj6hYjvLO7CzU1kV2K0g-VvaJZVPOusxzQiT58g14ref-RgmU6pLVfeqbiTkPE4SqHAik9cu6vRxXLlzu7gaHnigUIM742\" alt=\"sistemas lineares soma ou adi&ccedil;&atilde;o\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Vamos escolher nomes para as <span id=\"urn:enhancement-9471bf9f-845c-495c-9bba-d8cd355c87ff\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>, o que facilitar&aacute; o entendimento:<\/p><p class=\"wp-block-paragraph\"><strong>I &rarr; 3x + y = 40<\/strong><\/p><p class=\"wp-block-paragraph\"><strong>II &rarr; x + 2y = 20<\/strong><\/p><p class=\"wp-block-paragraph\">Como queremos desaparecer com uma das inc&oacute;gnitas, podemos multiplicar uma das <span id=\"urn:enhancement-b7c0d20c-580b-4597-b49e-8f26d34bdfc9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span>. Para isso, escolhemos um valor que facilite a elimina&ccedil;&atilde;o. No caso anterior, se multiplicarmos toda a <span id=\"urn:enhancement-c1edfdd9-1371-4536-ad4e-f0d560eb3cb7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> II por -3:<\/p><p class=\"wp-block-paragraph\">&nbsp;x + 2y = 20&nbsp;<br>-3.(x+2y) = -3.20<\/p><p class=\"wp-block-paragraph\"><strong><span id=\"urn:enhancement-5f71720f-3a61-46cb-8dce-fb6c81263953\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> II: -3x &ndash; 6y = -60&nbsp;<\/strong><\/p><p class=\"wp-block-paragraph\">Com essa nova <span id=\"urn:enhancement-04670946-f32c-4b03-aa19-8a73a5788013\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span>, fa&ccedil;amos a <span id=\"urn:enhancement-e3be0027-56c3-402e-8f7c-12755c6af95b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> entre as <span id=\"urn:enhancement-6036c92f-aebb-4aad-aa59-ce709c908e1a\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> I e II.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/xXlsmGo3za6FfdvqM9OEfMbAIrGrYO3j2TZ0DU6aEFYrLl9AFbiD3iXGPYV4Ircdil7wqeS5RtjO5qGVivRGyAAvDUXXeUnYMpsR4ZGV6lTHL9GsFphfR6_NtEA2D4C75y79mc2I\" alt=\"Soma \"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Perceba que a <span id=\"urn:enhancement-79accab2-7dac-47f6-ba13-42a1d5e3775d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> fornecida pela <span id=\"urn:enhancement-efa8754c-0523-4223-a7b0-3bbbdff525da\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> das <span id=\"urn:enhancement-1acbc415-f9ba-468a-81e9-f2a1472611e6\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> permite descobrir o valor de y:<\/p><p class=\"wp-block-paragraph\">0x &ndash; 5y = -20<br>-5y = -20<br>y = -5 \/- 20<br><strong>y = 4<\/strong><\/p><p class=\"wp-block-paragraph\">Seguimos com a substitui&ccedil;&atilde;o do valor de y=4 na <span id=\"urn:enhancement-d8169972-2c1f-45bb-8bab-58b81c6852bd\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> I:<\/p><p class=\"wp-block-paragraph\">3x + y = 40<br>3x + 4 = 40<br>3x = 36<br><strong>x = 12<\/strong><\/p><p class=\"wp-block-paragraph\"><strong>S = {12,4}<\/strong><\/p><h2 class=\"wp-block-heading\" id=\"h-estude-para-o-enem-com-o-estrategia-vestibulares\"><span class=\"ez-toc-section\" id=\"Estude-para-o-Enem-com-o-Estrategia-Vestibulares\"><\/span>Estude para o Enem com o Estrat&eacute;gia Vestibulares!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Vai fazer Enem e tem medo da <span id=\"urn:enhancement-cae3ce3e-81b4-46b3-ad52-ae3a7b76716b\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>? Conhe&ccedil;a o curso do Estrat&eacute;gia Vestibulares. As aulas s&atilde;o aprofundadas e todas as disciplinas focam nos assuntos de maior recorr&ecirc;ncia. Quer saber mais? Clique no banner abaixo.<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/pacote-enem\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/Enem.jpg\" alt=\"CTA curso Enem\" title=\"CTA curso Enem\">\n        <\/a>\n\n\n\n\n\n<\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb&eacute;m:<span class=\"ez-toc-section-end\"><\/span><\/h2><ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/funcao-de-primeiro-grau\/\">Fun&ccedil;&atilde;o de Primeiro Grau: o que &eacute; e como aplicar em prova?<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-do-primeiro-grau\/\">Equa&ccedil;&atilde;o do primeiro grau: o que &eacute;, tipos, como resolver e 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teorema?<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/regra-de-sinais\/\">Regra de sinais: o que &eacute; e como funciona o jogo de sinais<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-fuvest-2020\/\">Prova de matem&aacute;tica Fuvest 2020 com Resolu&ccedil;&atilde;o Comentada<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-2-fase-unicamp-2020\/\">Resolu&ccedil;&atilde;o da prova de Matem&aacute;tica da 2&ordf; Fase da Unicamp 2020<\/a>&nbsp;<\/li>\n\n\n\n<li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-unesp-2020\/\">Prova de Matem&aacute;tica UNESP 2020 com Resolu&ccedil;&atilde;o Comentada<\/a><\/li>\n<\/ul><\/p>\n","protected":false},"excerpt":{"rendered":"Imagine que voc&ecirc; possui duas equa&ccedil;&otilde;es, com duas inc&oacute;gnitas e a sua miss&atilde;o &eacute; descobrir o valor que&hellip;\n","protected":false},"author":23,"featured_media":68419,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-66322","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Sistemas lineares: o que s\u00e3o, como resolver e aplica\u00e7\u00f5es<\/title>\n<meta name=\"description\" content=\"Conhe\u00e7a os sistemas lineares: quando equa\u00e7\u00f5es s\u00e3o relacionadas para que sejam encontrados valores relacionados a cada inc\u00f3gnita.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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