{"id":25852,"date":"2023-05-21T12:37:14","date_gmt":"2023-05-21T11:37:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=25852"},"modified":"2023-05-21T15:50:15","modified_gmt":"2023-05-21T14:50:15","slug":"classificacao-de-sistemas-de-equacoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=25852","title":{"rendered":"Classifica\u00e7\u00e3o de sistemas de equa\u00e7\u00f5es"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_25852' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_25852' class='GTTabs_curr'><a  id=\"25852_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_25852' ><a  id=\"25852_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_25852'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considera os seguintes sistemas de equa\u00e7\u00f5es:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{{\\rm{(I)}}\\left\\{ {\\begin{array}{*{20}{l}}{2y &#8211; x = 0}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp;{{\\rm{(II)}}\\left\\{ {\\begin{array}{*{20}{l}}{x + y = &#8211; 4}\\\\{2x + 2y = &#8211; 8}\\end{array}} \\right.}&amp;{{\\rm{(III)}}\\left\\{ {\\begin{array}{*{20}{l}}{x + y = 36}\\\\{3x &#8211; y = 44}\\end{array}} \\right.}\\end{array}\\]<\/p>\n<ol>\n<li>Resolve graficamente os sistemas (I), (II) e (III).<\/li>\n<li>Resolve cada um dos sistemas de equa\u00e7\u00f5es, usando o m\u00e9todo de substitui\u00e7\u00e3o.<\/li>\n<li>Classifica os sistemas de equa\u00e7\u00f5es.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_25852' onClick='GTTabs_show(1,25852)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_25852'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considera os seguintes sistemas de equa\u00e7\u00f5es:<\/p>\n<\/blockquote>\n<p>\\[\\begin{array}{*{20}{c}}{{\\rm{(I)}}\\left\\{ {\\begin{array}{*{20}{l}}{2y &#8211; x = 0}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp;{{\\rm{(II)}}\\left\\{ {\\begin{array}{*{20}{l}}{x + y = &#8211; 4}\\\\{2x + 2y = &#8211; 8}\\end{array}} \\right.}&amp;{{\\rm{(III)}}\\left\\{ {\\begin{array}{*{20}{l}}{x + y = 36}\\\\{3x &#8211; y = 44}\\end{array}} \\right.}\\end{array}\\]<\/p>\n<p>\u00a0<\/p>\n<ol>\n<li>Em cada um dos sistemas, comecemos por resolver cada uma das equa\u00e7\u00f5es em ordem a y.<br \/><br \/>\n<table style=\"border-collapse: collapse;\">\n<tbody>\n<tr>\n<td style=\"vertical-align: top;\">(I)<\/td>\n<td>\n<p>\\(\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{2y &#8211; x = 0}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = \\frac{1}{2}x}\\\\{y = \\frac{1}{2}x + \\frac{1}{4}}\\end{array}} \\right.}\\end{array}\\)<\/p>\n<p>O sistema \u00e9 imposs\u00edvel, pois as duas fun\u00e7\u00f5es afins possuem gr\u00e1ficos que s\u00e3o retas estritamente paralelas, visto que possuem declives iguais (\\({\\frac{1}{2}}\\)) e ordenadas na origem diferentes: \\(0\\) e \\({\\frac{1}{4}}\\).<\/p>\n<\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25858\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25858\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1.png\" data-orig-size=\"924,463\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag201-T12-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1.png\" class=\"alignnone wp-image-25858\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1.png\" alt=\"\" width=\"420\" height=\"210\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1.png 924w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1-300x150.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1-768x385.png 768w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"vertical-align: top;\">(II)<\/td>\n<td>\n<p>\\(\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{x + y = &#8211; 4}\\\\{2x + 2y = &#8211; 8}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x &#8211; 4}\\\\{y = &#8211; x &#8211; 4}\\end{array}} \\right.}\\end{array}\\)<\/p>\n<p>O sistema \u00e9 poss\u00edvel e indeterminado, pois as duas fun\u00e7\u00f5es afins possuem gr\u00e1ficos que s\u00e3o retas paralelas coincidentes, visto que possuem declives iguais (\u20131) e ordenadas na origem iguais: \u20134.<\/p>\n<\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25859\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25859\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2.png\" data-orig-size=\"935,743\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag201-T12-2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2.png\" class=\"alignnone wp-image-25859\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2-300x238.png\" alt=\"\" width=\"420\" height=\"334\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2-300x238.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2-768x610.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-2.png 935w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"vertical-align: top;\">(III)<\/td>\n<td>\n<p>\\(\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{x + y = 36}\\\\{3x &#8211; y = 44}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x + 36}\\\\{y = 3x &#8211; 44}\\end{array}} \\right.}\\end{array}\\)<\/p>\n<p>O sistema \u00e9 poss\u00edvel e determinado, pois as duas fun\u00e7\u00f5es afins possuem gr\u00e1ficos que s\u00e3o retas concorrentes, visto que possuem declives diferentes: \u20131 e 3.<\/p>\n<\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25860\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25860\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3.png\" data-orig-size=\"920,733\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag201-T12-3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3.png\" class=\"wp-image-25860 alignnone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3-300x239.png\" alt=\"\" width=\"420\" height=\"335\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3-300x239.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3-768x612.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-3.png 920w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Resolvendo cada um dos sistemas de equa\u00e7\u00f5es, usando o m\u00e9todo de substitui\u00e7\u00e3o, vem:<br \/>\\[\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{2y &#8211; x = 0}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = \\frac{1}{2}x}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = \\frac{1}{2}x}\\\\{\\underbrace {2x &#8211; 2x = 1}_{{\\rm{Eq}}{\\rm{.\\,imposs\u00edvel}}}}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{x \\in \\emptyset \\wedge y \\in \\emptyset }\\end{array}\\]<br \/>\\[\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{x + y = &#8211; 4}\\\\{2x + 2y = &#8211; 8}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x &#8211; 4}\\\\{2x &#8211; 2x &#8211; 8 = &#8211; 8}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x &#8211; 4}\\\\{\\underbrace {0x = 0}_{{\\rm{Eq}}{\\rm{.\\,poss\u00edvel\\,indeterminada}}}}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{x \\in \\mathbb{R} \\wedge y = &#8211; x &#8211; 4}\\end{array}\\]<br \/>\\[\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{x + y = 36}\\\\{3x &#8211; y = 44}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x + 36}\\\\{3x + x &#8211; 36 = 44}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; x + 36}\\\\{4x = 80}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{x = 20}\\\\{y = 16}\\end{array}} \\right.}\\end{array}\\]<\/li>\n<li>A classifica\u00e7\u00e3o dos sistemas de equa\u00e7\u00f5es j\u00e1 foi feita em 1.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_25852' onClick='GTTabs_show(0,25852)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considera os seguintes sistemas de equa\u00e7\u00f5es: \\[\\begin{array}{*{20}{c}}{{\\rm{(I)}}\\left\\{ {\\begin{array}{*{20}{l}}{2y &#8211; x = 0}\\\\{4y &#8211; 2x = 1}\\end{array}} \\right.}&amp;{{\\rm{(II)}}\\left\\{ {\\begin{array}{*{20}{l}}{x + y = &#8211; 4}\\\\{2x + 2y = &#8211; 8}\\end{array}} \\right.}&amp;{{\\rm{(III)}}\\left\\{ {\\begin{array}{*{20}{l}}{x +&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":25862,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,714],"tags":[424,345,344,239],"series":[],"class_list":["post-25852","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-equacoes-literais-e-sistemas","tag-8-o-ano","tag-funcao-afim","tag-grafico","tag-sistema-de-equacoes"],"views":85,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag201-T12-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/25852","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=25852"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/25852\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/25862"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=25852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=25852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=25852"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=25852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}