{"id":33684,"date":"2021-04-20T13:00:13","date_gmt":"2021-04-20T13:00:13","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33684"},"modified":"2025-07-30T19:25:42","modified_gmt":"2025-07-30T19:25:42","slug":"aequatio-rectae-et-systemata-cartesianorum","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/","title":{"rendered":"Aequatio Rectae et Systemata Cartesianorum"},"content":{"rendered":"<p><center><\/p>\n<h1>Aequatio Rectae et Systemata Cartesianorum<\/h1>\n<p style=\"text-align: center;\"><em><strong>Summarium:<\/strong><br \/>\n      In hac lectione fundamenta geometriae analyticae tractabimus, ostendentes quomodo puncta in plano per coordinatas repraesentari possint et quomodo aequatio rectae ex declivitate et puncto dato formari queat. Explicantur notiones clavis, ut declivitas, usus aequationis <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span> et repraesentatio graphica linearum, praeterea exercitationes practicae atque applicationes adhibentur ad problemata in contextibus realibus solvenda, sicut positio computanda et intersectio inter rectas.<br \/>\n   <\/em><\/p>\n<p>   <strong>Proposita Discendi<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intellegere<\/strong> principia fundamentalia geometriae analyticae earumque applicationem in repraesentatione punctorum in plano cartesiano.<\/li>\n<li><strong>Agnotare<\/strong> formulam declivitatis rectae eiusque significationem geometricam.<\/li>\n<li><strong>Adhibere<\/strong> aequationem generalem rectae <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span> ad relationes lineares describendas.<\/li>\n<li><strong>Computare<\/strong> aequationem rectae ex puncto uno et declivitate data.<\/li>\n<li><strong>Graphice repraesentare<\/strong> rectas in plano cartesiano per earum aequationem linearim.<\/li>\n<li><strong>Solvere<\/strong> problemata quae intersectionem duarum rectarum per systemata aequationum involvunt.<\/li>\n<li><strong>Examinare<\/strong> relationem inter duas magnitudines lineares et quomodo hae per aequationem rectae repraesentari possint.<\/li>\n<\/ol>\n<p>   <strong>INDEX RERUM<\/strong><br \/>\n   <a href=\"#1\">Principia Geometriae Analyticae<\/a><br \/>\n   <a href=\"#2\">Aequatio Rectae<\/a><br \/>\n   <a href=\"#3\">Quomodo Graphice Repraesentetur Aequatio Rectae<\/a><br \/>\n   <a href=\"#4\">Intersectiones Inter Rectas<\/a>\n   <\/p>\n<p><\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mNISGHOByAI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">Nunc studium nostrum de aequatione rectae, systematibus cartesianis et principiis geometriae analyticae incipiemus.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Principia Geometriae Analyticae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><strong>Cum numeri reales introducuntur<\/strong><\/a>, plerumque dicitur hos esse puncta super rectam.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"RECTA DE LOS REALES\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"RECTA DE LOS REALES\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ex hoc, Cartesius ingeniose usus est duabus rectis ad puncta in plano repraesentanda per par coordinatarum (x,y).<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><\/noscript><\/p>\n<p><a name=\"2\"><\/a>     <\/p>\n<h2>Aequatio Rectae<\/h2>\n<p style=\"text-align: justify;\">His notionibus adhibitis, iam fieri potest considerare collectionem punctorum in plano ad curvas formandas, ubi coordinatae <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> respondet alia coordinata <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, et haec lex correspondentiae a functione determinatur. Hoc punctum est ubi algebra in geometriam ingreditur et nascitur \u00abGeometria Analytica\u00bb.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=320s\" target=\"_blank\" rel=\"noopener\"><strong>Geometrice rectam intellegimus<\/strong><\/a> ut curvam quae duos punctos iungit per viam brevissimam.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"RECTA EN EL PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"RECTA EN EL PLANO CARTESIANO\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Geometrice rectam intellegimus ut curvam quae duos punctos iungit per viam brevissimam. Hoc analysantes, secundum theorema Thaletis, videbimus quod omni incremento coordinatae <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> respondet incrementum coordinatae <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tale ut ratio <span class=\"katex-eq\" data-katex-display=\"false\">m=(y_2 - y_1)\/(x_2 - x_1)=\\Delta y \/ \\Delta x<\/span> semper sit constans pro quolibet par punctorum in recta. Hoc est quod <strong>\u00abdeclivitas rectae\u00bb<\/strong> appellatur.<\/p>\n<p style=\"text-align: justify;\">Cum declivitas sit eadem pro quolibet par punctorum in recta, si puncta rectae cum coordinatis <span class=\"katex-eq\" data-katex-display=\"false\">(x,y),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span> consideremus, scribere possumus:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{y-y_0}{x - x_0} = \\frac{y_2 - y_1}{x_2 - x_1}<\/span>\n<p style=\"text-align: justify;\">Quod idem est ac dicere<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{matrix}y &amp; = &amp; \\displaystyle \\frac{y_2 - y_1}{x_2 - x_1} (x - x_0 ) + y_0 \\\\ \\\\ &amp; = &amp; \\displaystyle \\frac{\\Delta y}{\\Delta x} (x - x_0) + y_0 \\end{matrix}<\/span>\n<p style=\"text-align: justify;\">Hinc oritur nota <strong>aequatio rectae<\/strong><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{red}{{y = m(x-x_0) + y_0}}<\/span>\n<p style=\"text-align: justify;\">Hic, par <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)<\/span> est punctum fixum, dum par <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)<\/span> est punctum quodlibet.<\/p>\n<h3>Exercitia Exemplaria<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Calcula aequationem rectae quae transit per punctum <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)=(2,3)<\/span> cum declivitate <span class=\"katex-eq\" data-katex-display=\"false\">m=3\/2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=695s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<li>Calcula aequationem rectae quae transit per punctum <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)=(1,8)<\/span> cum declivitate <span class=\"katex-eq\" data-katex-display=\"false\">m=7\/5<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=750s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<li>Calcula aequationem rectae quae transit per puncta <span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)=(3,5)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)=(1,-2)<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=818s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Quomodo Graphice Repraesentetur Aequatio Rectae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1063s\" target=\"_blank\" rel=\"noopener\"><strong>Iam vidimus quomodo obtineatur<\/strong><\/a> aequatio rectae ex informatione graphica; nunc viam contrariam sequemur, repraesentationem graphicam ex aequatione rectae obtinere.<\/p>\n<p style=\"text-align: justify;\">In fine diei, aequatio rectae semper exhibetur hoc modo:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y=mx + b<\/span>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">m=\\Delta Y \/ \\Delta x<\/span> est declivitas et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> est coefficient positio. Ex hoc oritur figura sequens:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"RECTA EN EL PLANO CARTESIANO CON COORDENADAS\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"RECTA EN EL PLANO CARTESIANO CON COORDENADAS\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><\/noscript><\/p>\n<h3>Exercitium Exemplare<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Repraesenta rectam cuius aequatio est <span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle \\frac{3}{4}x + 2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1139s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<li>Repraesenta rectam cuius aequatio est <span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle -\\frac{2}{5}x + 6<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1209s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<\/ol>\n<h3>Problemata Applicationis Aequationis Rectae<\/h3>\n<p style=\"text-align: justify;\">Recta adhiberi potest ad solvenda problemata quae relationem directam inter duas magnitudines implicant, ut in sequentibus exemplis:<\/p>\n<ol style=\"text-align: justify;\">\n<li>Vehiculum cum positione initiali <span class=\"katex-eq\" data-katex-display=\"false\">x_0 = 12[m]<\/span> movetur celeritate <span class=\"katex-eq\" data-katex-display=\"false\">v=0,3[m\/s]<\/span>; quae erit eius positio post <span class=\"katex-eq\" data-katex-display=\"false\">30[s]<\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1257s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<li>Homo ad mercatum it et emit <span class=\"katex-eq\" data-katex-display=\"false\">1[kg]<\/span> malorum, solvendo in summa <span class=\"katex-eq\" data-katex-display=\"false\">50 Z\\$.<\/span> Eodem die, idem homo iterum ad mercatum redit ut emat <span class=\"katex-eq\" data-katex-display=\"false\">3[kg]<\/span> malorum, solvendo in summa <span class=\"katex-eq\" data-katex-display=\"false\">60 Z\\$.<\/span>. Quaenam est pretia malorum et pretia vehiculorum? <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1383s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Intersectiones Inter Rectas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1855s\" target=\"_blank\" rel=\"noopener\"><strong>Supponamus duas rectas habere<\/strong><\/a> et volumus invenire punctum commune inter eas; hoc est, invenire intersectionem inter rectas. Ad huiusmodi problemata solvenda, necesse est systema aequationum resolvere. Ut hoc melius intellegamus, videamus exemplum sequentem.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1914s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremus rectas sequentes:<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_1 \\; : \\; y= \\displaystyle \\frac{3}{2}x + 1<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_2 \\; : \\; y=\\displaystyle -\\frac{1}{3}x + 9<\/span>\n<p style=\"text-align: justify;\">Ubi hae duae rectae intersecentur?<\/p>\n<p style=\"text-align: justify;\">Ad hoc resolvendum, ratiocinium sequens sequimur:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle \\frac{3}{2}x + 1<\/span><\/td>\n<td>; Recta <span class=\"katex-eq\" data-katex-display=\"false\">L_1<\/span><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; Recta <span class=\"katex-eq\" data-katex-display=\"false\">L_2<\/span><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x + 1 = -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; Ex (1) et (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x = -\\frac{1}{3}x + 8<\/span><\/td>\n<td>; Subtracto 1 ex utraque parte<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">9x = -2x + 48<\/span><\/td>\n<td>; Multiplicando utrumque latus per 6<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">11x =48<\/span><\/td>\n<td>; Addendo 2x ad utrumque latus<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x = \\frac{48}{11}<\/span><\/td>\n<td>; Dividendo utrumque latus per 11<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{2}\\cdot \\frac{48}{11} + 1<\/span><\/td>\n<td>; Ex (1) et (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{1}\\cdot \\frac{24}{11} + \\frac{11}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle \\frac{83}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x,y)= \\left(\\frac{48}{11}, \\frac{83}{11} \\right)<\/span><\/td>\n<td>; Ex (3) et (4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ergo, punctum intersectionis inter rectas est <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)= \\displaystyle \\left(\\frac{48}{11}, \\frac{83}{11} \\right).<\/span>\n<h3>Exemplum Problematum Applicationis ad Intersectionem Inter Rectas<\/h3>\n<p style=\"text-align: justify;\">Ad celebrationem, vendita sunt in summa 600 tesserae cum pecunia collecta totali <span class=\"katex-eq\" data-katex-display=\"false\">\\$1.300.000.<\/span>. Tesserae pro iuvenibus venditae sunt ad pretium <span class=\"katex-eq\" data-katex-display=\"false\">\\$1.000,<\/span>, et tesserae pro adultis ad <span class=\"katex-eq\" data-katex-display=\"false\">\\$3.000<\/span>. Quot iuvenes et quot adulti ad celebrationem venerunt? <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=2255s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLUTIO]<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aequatio Rectae et Systemata Cartesianorum Summarium: In hac lectione fundamenta geometriae analyticae tractabimus, ostendentes quomodo puncta in plano per coordinatas repraesentari possint et quomodo aequatio rectae ex declivitate et puncto dato formari queat. Explicantur notiones clavis, ut declivitas, usus aequationis et repraesentatio graphica linearum, praeterea exercitationes practicae atque applicationes adhibentur ad problemata in contextibus realibus [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28865,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33684","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Aequatio Rectae et Systemata Cartesianorum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Aequatio Rectae et Systemata Cartesianorum\" \/>\n<meta property=\"og:description\" content=\"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-20T13:00:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-30T19:25:42+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Aequatio Rectae et Systemata Cartesianorum\" \/>\n<meta name=\"twitter:description\" content=\"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Aequatio Rectae et Systemata Cartesianorum\",\"datePublished\":\"2021-04-20T13:00:13+00:00\",\"dateModified\":\"2025-07-30T19:25:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/\"},\"wordCount\":907,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/recta.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/\",\"name\":\"Aequatio Rectae et Systemata Cartesianorum - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/recta.jpg\",\"datePublished\":\"2021-04-20T13:00:13+00:00\",\"dateModified\":\"2025-07-30T19:25:42+00:00\",\"description\":\"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#primaryimage\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/recta.jpg\",\"contentUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/recta.jpg\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-rectae-et-systemata-cartesianorum\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Aequatio Rectae et Systemata Cartesianorum\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Aequatio Rectae et Systemata Cartesianorum - toposuranos.com\/material","description":"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/","og_locale":"es_ES","og_type":"article","og_title":"Aequatio Rectae et Systemata Cartesianorum","og_description":"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.","og_url":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-04-20T13:00:13+00:00","article_modified_time":"2025-07-30T19:25:42+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta-1024x585.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Aequatio Rectae et Systemata Cartesianorum","twitter_description":"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Aequatio Rectae et Systemata Cartesianorum","datePublished":"2021-04-20T13:00:13+00:00","dateModified":"2025-07-30T19:25:42+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/"},"wordCount":907,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg","articleSection":["Algebra et Geometria","Mathematica"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/","url":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/","name":"Aequatio Rectae et Systemata Cartesianorum - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg","datePublished":"2021-04-20T13:00:13+00:00","dateModified":"2025-07-30T19:25:42+00:00","description":"Disce fundamenta Aequationis Rectae et Systematum Cartesianorum. Cognosce quomodo rectas computare et graphice repraesentare, problemata solvere.","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/recta.jpg","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/la\/aequatio-rectae-et-systemata-cartesianorum\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Aequatio Rectae et Systemata Cartesianorum"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/33684","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=33684"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/33684\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/28865"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=33684"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=33684"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=33684"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}