{"id":6314,"date":"2010-12-01T22:18:49","date_gmt":"2010-12-01T22:18:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6314"},"modified":"2022-01-12T14:12:34","modified_gmt":"2022-01-12T14:12:34","slug":"resolva-classifique-e-interprete-geometricamente-as-solucoes-dos-seguintes-sistemas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6314","title":{"rendered":"Resolva, classifique e interprete geometricamente as solu\u00e7\u00f5es dos seguintes sistemas"},"content":{"rendered":"<p><ul id='GTTabs_ul_6314' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6314' class='GTTabs_curr'><a  id=\"6314_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6314' ><a  id=\"6314_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_6314'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span> Resolva, classifique e interprete geometricamente as solu\u00e7\u00f5es dos seguintes sistemas:<\/p>\n<ol>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y=0\u00a0 \\\\ \u00a0\u00a0 x+y+z=3\u00a0 \\\\ \u00a0\u00a0 x-z=1\u00a0 \\\\ \\end{array} \\right.$<br \/>\n\u00ad<\/li>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a-b-c=3\u00a0 \\\\ \u00a0\u00a0 2a-b+2c=2\u00a0 \\\\ \u00a0\u00a0 a+10b-3c=5\u00a0 \\\\ \\end{array} \\right.$<br \/>\n\u00ad<\/li>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2a-3b-2c=2\u00a0 \\\\ \u00a0\u00a0 4a-3b+c=4\u00a0 \\\\ \u00a0\u00a0 2a+12b-7c=2\u00a0 \\\\ \\end{array} \\right.$<br \/>\n\u00ad<\/li>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2y+z-x=0\u00a0 \\\\ \u00a0\u00a0 x+y-2z=5\u00a0 \\\\ \u00a0\u00a0 x+\\frac{3}{2}y-\\frac{1}{2}z=\\frac{15}{2}\u00a0 \\\\ \\end{array} \\right.$<br \/>\n\u00ad<\/li>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 4x-3y+z=4\u00a0 \\\\ \u00a0\u00a0 2x+3y-2z=2\u00a0 \\\\ \u00a0\u00a0 2x+12y-7z=2\u00a0 \\\\ \\end{array} \\right.$<br \/>\n\u00ad<\/li>\n<li>$\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+2y-6z=4\u00a0 \\\\ \u00a0\u00a0 2x-2y+3z=4\u00a0 \\\\ \u00a0\u00a0 x+8y-21z=6\u00a0 \\\\ \\end{array} \\right.$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6314' onClick='GTTabs_show(1,6314)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6314'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 -1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \u00a0\u00a0 {}\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y=0\u00a0 \\\\ \u00a0\u00a0 x+y+z=3\u00a0 \\\\ \u00a0\u00a0 x-z=1\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y=0\u00a0 \\\\ \u00a0\u00a0 z=3\u00a0 \\\\ \u00a0\u00a0 x-z=1\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 y=-4\u00a0 \\\\ \u00a0\u00a0 z=3\u00a0 \\\\ \u00a0\u00a0 x=4\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}\\]<br \/>\nO sistema \u00e9 poss\u00edvel e determinado.<br \/>\nA intersec\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 o ponto de coordenadas $(4,-4,3)$.<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \u00a0\u00a0 -2\\times\u00a0\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a-b-c=3\u00a0 \\\\ \u00a0\u00a0 2a-b+2c=2\u00a0 \\\\ \u00a0\u00a0 a+10b-3c=5\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a-b-c=3\u00a0 \\\\ \u00a0\u00a0 b+4c=-4\u00a0 \\\\ \u00a0\u00a0 -21b+8c=-8\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a-b-c=3\u00a0 \\\\ \u00a0\u00a0 b+4c=-4\u00a0 \\\\ \u00a0\u00a0 -23b=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a=2\u00a0 \\\\ \u00a0\u00a0 c=-1\u00a0 \\\\ \u00a0\u00a0 b=0\u00a0 \\\\ \\end{array} \\right. &amp; {} &amp; {} &amp; {}\u00a0 \\\\ \\end{array}\\]<br \/>\nO sistema \u00e9 poss\u00edvel e determinado.<br \/>\nA intersec\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 o ponto de coordenadas $(2,0,-1)$.<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 -1\\times\u00a0 &amp; -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; +\u00a0 \\\\ \u00a0\u00a0 + &amp; {}\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2a-3b-2c=2\u00a0 \\\\ \u00a0\u00a0 4a-3b+c=4\u00a0 \\\\ \u00a0\u00a0 2a+12b-7c=2\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2a-3b-2c=2\u00a0 \\\\ \u00a0\u00a0 3b+5c=0\u00a0 \\\\ \u00a0\u00a0 15b-5c=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2a-3b-2c=2\u00a0 \\\\ \u00a0\u00a0 3b+5c=0\u00a0 \\\\ \u00a0\u00a0 18b=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a=1\u00a0 \\\\ \u00a0\u00a0 c=0\u00a0 \\\\ \u00a0\u00a0 b=0\u00a0 \\\\ \\end{array} \\right. &amp; {} &amp; {} &amp; {}\u00a0 \\\\ \\end{array}\\]<br \/>\nO sistema \u00e9 poss\u00edvel e determinado.<br \/>\nA intersec\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 o ponto de coordenadas $(1,0,0)$.<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2y+z-x=0\u00a0 \\\\ \u00a0\u00a0 x+y-2z=5\u00a0 \\\\ \u00a0\u00a0 x+\\frac{3}{2}y-\\frac{1}{2}z=\\frac{15}{2}\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 2\\times\u00a0 &amp; 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; +\u00a0 \\\\ \u00a0\u00a0 + &amp; {}\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 -x+2y+z=0\u00a0 \\\\ \u00a0\u00a0 x+y-2z=5\u00a0 \\\\ \u00a0\u00a0 2x+3y-z=15\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y-2z=5\u00a0 \\\\ \u00a0\u00a0 3y-z=5\u00a0 \\\\ \u00a0\u00a0 7y+z=15\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y-2z=5\u00a0 \\\\ \u00a0\u00a0 3y-z=5\u00a0 \\\\ \u00a0\u00a0 10y=20\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x=5\u00a0 \\\\ \u00a0\u00a0 z=1\u00a0 \\\\ \u00a0\u00a0 y=2\u00a0 \\\\ \\end{array} \\right. &amp; {}\u00a0 \\\\ \\end{array}\\]<br \/>\nO sistema \u00e9 poss\u00edvel e determinado.<br \/>\nA intersec\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 o ponto de coordenadas $(5,2,1)$.<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 {} &amp; +\u00a0 \\\\ \u00a0\u00a0 -1\\times\u00a0 &amp; -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 + &amp; {}\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 4x-3y+z=4\u00a0 \\\\ \u00a0\u00a0 2x+3y-2z=2\u00a0 \\\\ \u00a0\u00a0 2x+12y-7z=2\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 4x-3y+z=4\u00a0 \\\\ \u00a0\u00a0 -9y+5z=0\u00a0 \\\\ \u00a0\u00a0 9y-5z=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 4x-3y+z=4\u00a0 \\\\ \u00a0\u00a0 -9y+5z=0\u00a0 \\\\ \u00a0\u00a0 0y+0z=0\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}\\]<br \/>\nA equa\u00e7\u00e3o $0y+0z=0$ \u00e9 poss\u00edvel e indeterminada, o que significa que podemos atribuir a y ou a z um qualquer valor real. Seja $z=k\\,,\\,\\,k\\in \\mathbb{R}$.<br \/>\nO sistema anterior pode ent\u00e3o ser substitu\u00eddo por:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 z=k\u00a0 \\\\ \u00a0\u00a0 -9y+5z=0\u00a0 \\\\ \u00a0\u00a0 4x-3y+z=4\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 z=k\u00a0 \\\\ \u00a0\u00a0 y=\\frac{5}{9}k\u00a0 \\\\ \u00a0\u00a0 x=\\frac{4-k+\\frac{5}{3}k}{4}\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 z=k\u00a0 \\\\ \u00a0\u00a0 y=\\frac{5}{9}k\u00a0 \\\\ \u00a0\u00a0 x=1+\\frac{1}{6}k\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}\\]<br \/>\nO sistema \u00e9 poss\u00edvel e indeterminado, pois admite como solu\u00e7\u00e3o qualquer terno ordenado do tipo $(1+\\frac{1}{6}k,\\frac{5}{9}k,k)\\,,\\,\\,k\\in \\mathbb{R}$.<br \/>\nA intersec\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 uma recta, que pode ser definida vectorialmente por: \\[(x,y,z)=(1,0,0)+k(\\frac{1}{6},\\frac{5}{9},1)\\,,\\,\\,k\\in \\mathbb{R}\\]<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 -1\\times\u00a0 &amp; -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; +\u00a0 \\\\ \u00a0\u00a0 + &amp; {}\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+2y-6z=4\u00a0 \\\\ \u00a0\u00a0 2x-2y+3z=4\u00a0 \\\\ \u00a0\u00a0 x+8y-21z=6\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+2y-6z=4\u00a0 \\\\ \u00a0\u00a0 -6y+15z=-4\u00a0 \\\\ \u00a0\u00a0 6y-15z=2\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+2y-6z=4\u00a0 \\\\ \u00a0\u00a0 6y-15z=2\u00a0 \\\\ \u00a0\u00a0 0y+0z=-2\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}\\]<br \/>\nA equa\u00e7\u00e3o $0y+0z=-2$ \u00e9 imposs\u00edvel, pelo que o sistema \u00e9 tamb\u00e9m imposs\u00edvel.<br \/>\nLogo, o seu conjunto-solu\u00e7\u00e3o \u00e9 $S=\\left\\{ {} \\right\\}$.<br \/>\nA interse\u00e7\u00e3o dos planos definidos pelas tr\u00eas equa\u00e7\u00f5es \u00e9 um conjunto vazio.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6314' onClick='GTTabs_show(0,6314)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Resolva, classifique e interprete geometricamente as solu\u00e7\u00f5es dos seguintes sistemas: $\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y=0\u00a0 \\\\ \u00a0\u00a0 x+y+z=3\u00a0 \\\\ \u00a0\u00a0 x-z=1\u00a0 \\\\ \\end{array} \\right.$ \u00ad $\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a-b-c=3\u00a0 \\\\ \u00a0\u00a0 2a-b+2c=2\u00a0&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19407,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,110],"tags":[422,67,119],"series":[],"class_list":["post-6314","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-interseccao-de-planos"],"views":3911,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Intersecao_de_tres_planos.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6314","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=6314"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6314\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19407"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6314"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6314"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6314"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}