{"id":25676,"date":"2023-05-15T21:53:38","date_gmt":"2023-05-15T20:53:38","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=25676"},"modified":"2023-05-16T23:37:03","modified_gmt":"2023-05-16T22:37:03","slug":"o-dinheiro-da-salome-e-da-ines","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=25676","title":{"rendered":"O dinheiro da Salom\u00e9 e da In\u00eas"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_25676' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_25676' class='GTTabs_curr'><a  id=\"25676_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_25676' ><a  id=\"25676_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_25676'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25679\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25679\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\" data-orig-size=\"906,436\" 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_Pag196-T7\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\" class=\"aligncenter wp-image-25679\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-300x144.png\" alt=\"\" width=\"720\" height=\"346\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-300x144.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-768x370.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png 906w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/a><\/p>\n<ol>\n<li>Designando por <em>x<\/em> o dinheiro que a Salom\u00e9 tem na carteira e por <em>y<\/em> o da In\u00eas, traduz as afirma\u00e7\u00f5es atrav\u00e9s de duas equa\u00e7\u00f5es.<\/li>\n<li>Representa, no mesmo referencial, as fun\u00e7\u00f5es afins definidas por \\(y = &#8211; \\frac{1}{2}x + 10\\) e \\(y = 2x &#8211; 10\\), e marca o ponto <em>I<\/em> que corresponde \u00e0 sua interse\u00e7\u00e3o.<\/li>\n<li>Indica, a partir das representa\u00e7\u00f5es gr\u00e1ficas anteriores, quais s\u00e3o as coordenadas do ponto <em>I<\/em>.<br \/>O que representam essas coordenadas?<\/li>\n<li>H\u00e1 alguma rela\u00e7\u00e3o entre o que a Salom\u00e9 e a In\u00eas disseram e os gr\u00e1ficos que desenhaste?<br \/>Explica a tua resposta.<\/li>\n<li>Que dinheiro tem a In\u00eas? E a Salom\u00e9?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_25676' onClick='GTTabs_show(1,25676)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_25676'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25679\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25679\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\" data-orig-size=\"906,436\" 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_Pag196-T7\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png\" class=\"aligncenter wp-image-25679\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-300x144.png\" alt=\"\" width=\"720\" height=\"346\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-300x144.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7-768x370.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7.png 906w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/a><\/p>\n<ol>\n<li>Designando por:<br \/><em><strong>x<\/strong>:<\/em> o dinheiro (em euros) que a Salom\u00e9 tem na carteira;<br \/><em><strong>y<\/strong>:<\/em> o dinheiro (em euros) da In\u00eas;<br \/>podemos traduz as suas afirma\u00e7\u00f5es atrav\u00e9s das duas equa\u00e7\u00f5es seguintes:<br \/>\n<table class=\" aligncenter\" style=\"width: 90%; border-collapse: collapse;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Jovem<\/strong><\/td>\n<td style=\"width: 33.3333%;\"><strong>Afirma\u00e7\u00e3o<\/strong><\/td>\n<td style=\"width: 33.3333%;\"><strong>Equa\u00e7\u00e3o<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Salom\u00e9<\/td>\n<td style=\"width: 33.3333%;\">Se adicionar o dinheiro que tenho na carteira ao dobro do que a In\u00eas possui na dela, obtenho 20 \u20ac.<\/td>\n<td style=\"width: 33.3333%;\">\\(x + 2y = 20\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">In\u00eas<\/td>\n<td style=\"width: 33.3333%;\">Se retirar o dinheiro que tenho ao dobro do que tem a Salom\u00e9, fico com 10 \u20ac.<\/td>\n<td style=\"width: 33.3333%;\">\\(2x &#8211; y = 10\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>No referencial abaixo, est\u00e3o representadas as fun\u00e7\u00f5es afins definidas por \\(y = &#8211; \\frac{1}{2}x + 10\\) e \\(y = 2x &#8211; 10\\), bem como est\u00e1 marcado o ponto <em>I<\/em> que corresponde \u00e0 sua interse\u00e7\u00e3o.<br \/>\\[\\begin{array}{*{20}{c}}{\\begin{array}{*{20}{c}}x&amp;{y = &#8211; \\frac{1}{2}x + 10}&amp;{{\\rm{Ponto}}}\\\\\\hline0&amp;{10}&amp;{A\\left( {0,10} \\right)}\\\\{20}&amp;0&amp;{B\\left( {20,0} \\right)}\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{c}}x&amp;{y = 2x &#8211; 10}&amp;{{\\rm{Ponto}}}\\\\\\hline0&amp;{ &#8211; 10}&amp;{C\\left( {0, &#8211; 10} \\right)}\\\\{10}&amp;{10}&amp;{D\\left( {10,10} \\right)}\\end{array}}\\end{array}\\]<br \/><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"25698\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=25698\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b.png\" data-orig-size=\"1556,1052\" 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_Pag196-T7_grafico_b\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-1024x692.png\" class=\"aligncenter wp-image-25698\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-300x203.png\" alt=\"\" width=\"720\" height=\"487\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-1024x692.png 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-768x519.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b-1536x1038.png 1536w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_grafico_b.png 1556w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/a><\/li>\n<li>De acordo com as representa\u00e7\u00f5es gr\u00e1ficas anteriores, as coordenadas do ponto <em>I<\/em>\u00a0s\u00e3o \\(\\left( {8,6} \\right)\\).<br \/>Essas coordenadas representam a solu\u00e7\u00e3o comum das equa\u00e7\u00f5es das duas retas, o que est\u00e1 confirmado seguidamente, pela obten\u00e7\u00e3o de uma afirma\u00e7\u00e3o verdadeira resultante de se ter substitu\u00eddo em ambas as equa\u00e7\u00f5es as vari\u00e1veis <em>x<\/em> e <em>y<\/em> pelas coordenadas do ponto <em>I<\/em>:<br \/>\\[\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{6 = &#8211; \\frac{1}{2} \\times 8 + 10}\\\\{6 = 2 \\times 8 &#8211; 10}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{6 = 6}\\\\{6 = 6}\\end{array}} \\right.}\\end{array}\\]<\/li>\n<li>Sim, h\u00e1 uma rela\u00e7\u00e3o entre o que a Salom\u00e9 e a In\u00eas disseram e os gr\u00e1ficos que foram desenhados acima. Com efeito, as duas equa\u00e7\u00f5es resultantes das afirma\u00e7\u00f5es das jovens s\u00e3o equivalentes \u00e0s equa\u00e7\u00f5es utilizadas na representa\u00e7\u00e3o gr\u00e1fica conjunta:<br \/>\\[\\begin{array}{*{20}{c}}{\\left\\{ {\\begin{array}{*{20}{l}}{x + 2y = 20}\\\\{2x &#8211; y = 10}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{2y = &#8211; x + 20}\\\\{ &#8211; y = &#8211; 2x + 10}\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}{y = &#8211; \\frac{1}{2}x + 10}\\\\{y = 2x &#8211; 10}\\end{array}} \\right.}\\end{array}\\]<\/li>\n<li>A In\u00eas tem 6 \u20ac e a Salom\u00e9 tem 8 \u20ac.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_25676' onClick='GTTabs_show(0,25676)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Designando por x o dinheiro que a Salom\u00e9 tem na carteira e por y o da In\u00eas, traduz as afirma\u00e7\u00f5es atrav\u00e9s de duas equa\u00e7\u00f5es. Representa, no mesmo referencial, as fun\u00e7\u00f5es afins&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":25680,"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-25676","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":92,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/05\/8_Pag196-T7_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/25676","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=25676"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/25676\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/25680"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=25676"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=25676"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=25676"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=25676"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}