{"id":185577,"date":"2023-04-04T00:43:50","date_gmt":"2023-04-04T03:43:50","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=185577"},"modified":"2023-05-21T00:33:48","modified_gmt":"2023-05-21T03:33:48","slug":"problema-um-grande-sistema","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-um-grande-sistema\/","title":{"rendered":".Problem\u00e3o: Um grande sistema"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=189424\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#778899;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<div class=\"g1n\" style=\"line-height:25px;padding-bottom:15px\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<span style=\"font-size:12px\">(<strong>IME<\/strong>)<\/span> Considere o seguinte sistema de equa\u00e7\u00f5es lineares:<br \/>\n<center>[tex]\\begin{cases} 6x_{1} +x_{2}+x_{3}+x_{4}+x_{5}=10 \\\\x_{1} +6x_{2}+x_{3}+x_{4}+x_{5}=20 \\\\x_{1} +x_{2}+6x_{3}+x_{4}+x_{5}=40\\\\x_{1} +x_{2}+x_{3}+6x_{4}+x_{5}=80 \\\\x_{1} +x_{2}+x_{3}+x_{4}+6x_{5}=160 \\end{cases}[\/tex].<\/center><br \/>\nCalcule o valor de [tex]7x_{1}+3x_{5}[\/tex].\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:28px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 1px; margin-bottom: 20px;border: 0 none\"\/>\n<ul>\n<li>Somando todas  a equa\u00e7\u00f5es do sistema, obtemos:<br \/>\n[tex]\\qquad 10x_{1} +10x_{2}+10x_{3}+10x_{4}+10x_{5}=310[\/tex].<\/li>\n<li>Dividindo a equa\u00e7\u00e3o obtida  por [tex]10[\/tex], temos:<br \/>\n[tex]\\qquad x_{1} +x_{2}+x_{3}+x_{4}+x_{5}=31[\/tex].<\/li>\n<li>Voltando ao sistema, a equa\u00e7\u00e3o [tex]6x_{1} +x_{2}+x_{3}+x_{4}+x_{5}=10[\/tex] pode ser reescrita da seguinte forma:<br \/>\n[tex]\\qquad 5x_{1}+x_{1} +x_{2}+x_{3}+x_{4}+x_{5}=10[\/tex].<\/li>\n<li>Substituindo [tex]x_{1} +x_{2}+x_{3}+x_{4}+x_{5}[\/tex] por [tex]31[\/tex] na equa\u00e7\u00e3o obtida, segue que [tex]5x_{1}+31=10 [\/tex], ou seja, [tex]\\boxed{x_{1}=\\dfrac{-21}{5}}[\/tex].<\/li>\n<li>Analogamente, a equa\u00e7\u00e3o [tex]x_{1} +x_{2}+x_{3}+x_{4}+6x_{5}=160[\/tex] pode ser reescrita como:<br \/>\n[tex]\\qquad x_{1} +x_{2}+x_{3}+x_{4}+x_{5}+5x_{5}=160[\/tex].<\/li>\n<li>Substituindo [tex]x_{1} +x_{2}+x_{3}+x_{4}+x_{5}[\/tex] por [tex]31[\/tex] na equa\u00e7\u00e3o obtida, conclu\u00edmos que [tex]31+5x_{5}=160[\/tex], ou seja [tex]\\boxed{x_{5}=\\dfrac{129}{5}}[\/tex].<\/li>\n<\/ul>\n<p>Portanto,<br \/>\n [tex]\\qquad 7x_{1}+3x_{5}=7 \\cdot  \\left( \\dfrac{-21}{5} \\right)+ 3 \\cdot  \\left( \\dfrac{129}{5} \\right)\\\\<br \/>\n\\qquad 7x_{1}+3x_{5}=\\dfrac{-147+387}{5}=\\dfrac{240}{5}\\\\<br \/>\n\\qquad \\fcolorbox{black}{#eee0e5}{$7x_{1}+3x_{5}=48$}\\,.[\/tex]<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" >Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (Indicado a partir do 9\u00ba ano do E. F.) (IME) Considere o seguinte sistema de equa\u00e7\u00f5es lineares: [tex]\\begin{cases} 6x_{1} +x_{2}+x_{3}+x_{4}+x_{5}=10 \\\\x_{1} +6x_{2}+x_{3}+x_{4}+x_{5}=20 \\\\x_{1} +x_{2}+6x_{3}+x_{4}+x_{5}=40\\\\x_{1} +x_{2}+x_{3}+6x_{4}+x_{5}=80 \\\\x_{1} +x_{2}+x_{3}+x_{4}+6x_{5}=160 \\end{cases}[\/tex]. Calcule o valor de [tex]7x_{1}+3x_{5}[\/tex]. Solu\u00e7\u00e3o Somando todas a equa\u00e7\u00f5es do sistema, obtemos: [tex]\\qquad 10x_{1} +10x_{2}+10x_{3}+10x_{4}+10x_{5}=310[\/tex]. Dividindo a &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-um-grande-sistema\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-185577","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/185577","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=185577"}],"version-history":[{"count":9,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/185577\/revisions"}],"predecessor-version":[{"id":191196,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/185577\/revisions\/191196"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=185577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}