{"id":1976,"date":"2021-12-02T21:01:41","date_gmt":"2021-12-02T21:01:41","guid":{"rendered":"https:\/\/blogs.ugto.mx\/rea\/?p=1976"},"modified":"2023-10-03T17:27:58","modified_gmt":"2023-10-03T17:27:58","slug":"clase-digital-9-metodos-de-solucion-de-sistemas-de-ecuaciones-cuadraticas-con-dos-variables","status":"publish","type":"post","link":"https:\/\/blogs.ugto.mx\/rea\/clase-digital-9-metodos-de-solucion-de-sistemas-de-ecuaciones-cuadraticas-con-dos-variables\/","title":{"rendered":"Clase digital 9: M\u00e9todos de soluci\u00f3n de sistemas de ecuaciones cuadr\u00e1ticas con dos variables"},"content":{"rendered":"\n\n\n<div class=\"wp-block-cover is-light\" style=\"min-height:284px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"wp-block-cover__background has-background-dim-40 has-background-dim\"><\/span><img decoding=\"async\" class=\"wp-block-cover__image-background wp-image-1977\" alt=\"\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu.jpg\" style=\"object-position:41% 77%\" data-object-fit=\"cover\" data-object-position=\"41% 77%\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"1067\" class=\"wp-block-cover__image-background wp-image-1977\" alt=\"\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu.jpg\" style=\"object-position:41% 77%\" data-object-fit=\"cover\" data-object-position=\"41% 77%\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu.jpg 1600w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu-300x200.jpg 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu-1024x683.jpg 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu-768x512.jpg 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu-1536x1024.jpg 1536w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/djwvzy4pkgu-272x182.jpg 272w\" sizes=\"auto, (max-width: 1600px) 100vw, 1600px\" \/><\/noscript><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-base-3-color has-text-color has-large-font-size wp-block-paragraph\"><br>M\u00e9todos de soluci\u00f3n de sistemas de <br>ecuaciones cuadr\u00e1ticas con dos variables<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"introduccion\">Introducci\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Te doy la bienvenida a tu \u00faltima clase del curso de \u00c1lgebra II titulada \u201cM\u00e9todos de soluci\u00f3n de sistemas de ecuaciones cuadr\u00e1ticas con dos variables\u201d. Recuerda como lo vimos en la clase 7 y 8, un sistema de ecuaciones es un conjunto de dos o m\u00e1s ecuaciones que comparten dos o m\u00e1s inc\u00f3gnitas, donde la soluci\u00f3n satisface todas las ecuaciones del sistema.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ahora los sistemas de ecuaciones son cuadr\u00e1ticos, es decir, aquella que aparece al menos una ecuaci\u00f3n de orden dos. Y para su resoluci\u00f3n se usan las mismas t\u00e9cnicas aplicadas en los sistemas de ecuaciones de primer grado o lineales.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Primero analizaremos la t\u00e9cnica anal\u00edtica de resoluci\u00f3n de sistemas de ecuaciones cuadr\u00e1tica \u2013 lineal por m\u00e9todo de suma y resta. Posteriormente, veremos la resoluci\u00f3n de sistemas de ecuaciones cuadr\u00e1tica \u2013 cuadr\u00e1tica por m\u00e9todo algebraico. Finalmente resolveremos problemas cotidianos que se resuelvan mediante un modelo matem\u00e1tico que incluya un sistema cuadr\u00e1tico.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Disfruta tu \u00faltima sesi\u00f3n, comencemos.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"desarrollo-del-tema\">Desarrollo del tema<\/h2>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"1-1-sistema-con-una-ecuacion-lineal-y-una-cuadratica\">1.1. Sistema con una ecuaci\u00f3n lineal y una cuadr\u00e1tica<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">El m\u00e9todo id\u00f3neo para la soluci\u00f3n de este sistema es el m\u00e9todo de sustituci\u00f3n, como recordar\u00e1s, este m\u00e9todo utiliza la propiedad aditiva, multiplicativa y de sustituci\u00f3n; el procedimiento consiste en que despejes una de las variables de la ecuaci\u00f3n lineal y luego sustituyas dicha variable en la ecuaci\u00f3n cuadr\u00e1tica.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo 1: Analiza y asimila la soluci\u00f3n del sistema de ecuaciones: &nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2212 4x\u22126y+8 = 0<br>3x\u2212 y\u22128 = 0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Identifica que la ecuaci\u00f3n lineal es 3x \u2212 y \u22128 = 0, y despeja la variable &nbsp;\u00aby\u00bb, 3x \u22128 = y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Despu\u00e9s sustituye &nbsp;y = 3x \u22128 en la ecuaci\u00f3n cuadr\u00e1tica x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2212 4x \u2212 6y +8 = 0<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x<sup>2<\/sup>&nbsp;+(3x \u22128)<sup>2<\/sup>&nbsp;\u2212 4x \u2212 6(3x \u22128)+8 = 0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Desarrolla las operaciones: x<sup>2<\/sup>&nbsp;+ 9x<sup>2<\/sup>&nbsp;\u2212 48x + 64 \u2212 4x \u221218x + 48 +8 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suma los t\u00e9rminos semejantes: 10x<sup>2<\/sup>&nbsp;\u2212 70x +120 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Simplifica la ecuaci\u00f3n: x<sup>2<\/sup>&nbsp;\u2212 7x +12 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Factoriza: (x \u2212 3)(x \u2212 4)= 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n: x = 3, x = 4<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituye la soluci\u00f3n de la variable \u00abx\u00bb en la ecuaci\u00f3n y = 3x \u22128<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Concluye que la soluci\u00f3n es:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x=3, y=1<br>x=4, y=4<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">o bien los puntos (3,1) y (4,4)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo 2: Analiza y asimila la soluci\u00f3n (intersecciones de sus gr\u00e1ficas) del sistema de ecuaciones. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x \u2212 y \u2212 2 = 0<br>xy \u22128 = 0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Identifica que la ecuaci\u00f3n lineal es x \u2212 y \u2212 2 = 0 , y despeja la variable \u201cy\u201d: x \u2212 2 = y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituye&nbsp;y = x \u2212 2 en la ecuaci\u00f3n cuadr\u00e1tica xy \u22128 = 0<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x(x \u2212 2)\u22128 = 0<br>x<sup>2<\/sup>&nbsp;\u2212 2x \u22128 = (x + 2)(x \u2212 4)= 0<br>x =\u22122, x = 4<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n :(\u2212 2,\u22124), (4,2)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo3: Analiza la soluci\u00f3n del sistema de ecuaciones.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x+y+1=0<br>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ 6y -x = 7<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Identifica que la ecuaci\u00f3n lineal es x+y+1 = 0, &nbsp;despeja la variable \u00abx\u00bb: x=-1-y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;Luego, sustituye la variable \u00abx\u00bb en la ecuaci\u00f3n cuadr\u00e1tica x<sup>2<\/sup>&nbsp;+y<sup>2<\/sup>&nbsp;+6y -x =7 , y desarrolla las operaciones:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.13.png\" alt=\"\" class=\"wp-image-1978\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"335\" height=\"136\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.13.png\" alt=\"\" class=\"wp-image-1978\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.13.png 335w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.13-300x122.png 300w\" sizes=\"auto, (max-width: 335px) 100vw, 335px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Ahora, factoriza la ecuaci\u00f3n anterior: (2y\u22121)(y+ 5) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De cada factor concluye que:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.39.png\" alt=\"\" class=\"wp-image-1979\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"162\" height=\"83\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.23.39.png\" alt=\"\" class=\"wp-image-1979\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Despu\u00e9s, sustituye el valor de la variable \u00aby\u00bb en la ecuaci\u00f3n lineal x =-1 -y.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.24.03.png\" alt=\"\" class=\"wp-image-1980\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"776\" height=\"74\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.24.03.png\" alt=\"\" class=\"wp-image-1980\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.24.03.png 776w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.24.03-300x29.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.24.03-768x73.png 768w\" sizes=\"auto, (max-width: 776px) 100vw, 776px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Como siguiente actividad, te invito a analizar los siguientes videos en donde se explica a detalle c\u00f3mo resolver sistemas de ecuaciones cuadr\u00e1tica \u2013 lineal con m\u00e9todos algebraicos y gr\u00e1ficos.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/es.khanacademy.org\/math\/algebra2\/x2ec2f6f830c9fb89:eq\/x2ec2f6f830c9fb89:quad-sys\/v\/systems-of-nonlinear-equations-3\" target=\"_blank\" rel=\"noreferrer noopener\">Sistemas cuadr\u00e1ticos: una recta y un c\u00edrculo.<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/es.khanacademy.org\/math\/algebra2\/x2ec2f6f830c9fb89:eq\/x2ec2f6f830c9fb89:quad-sys\/v\/systems-of-nonlinear-equations-3\" target=\"_blank\" rel=\"noreferrer noopener\">Sistemas cuadr\u00e1ticos: soluci\u00f3n gr\u00e1fica<\/a>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"1-2-sistema-formado-con-dos-ecuaciones-cuadraticas\">1.2. Sistema formado con dos ecuaciones cuadr\u00e1ticas.&nbsp;<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Para este sistema, utiliza el m\u00e9todo de eliminaci\u00f3n (suma o resta) y el de sustituci\u00f3n. En el m\u00e9todo de suma o resta, utiliza la propiedad multiplicativa y aditiva, para que elimines una de las variables, multiplicando las ecuaciones por n\u00fameros que hagan posible la anulaci\u00f3n. Enseguida, analiza los siguientes casos:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Caso 1: Sistema con dos ecuaciones de la forma <em>ax<sup>2<\/sup>&nbsp;+by<sup>2<\/sup>&nbsp;+ cx+ dy + e = 0<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En estos sistemas, es conveniente que elimines por medio de la propiedad aditiva uno de los t\u00e9rminos cuadr\u00e1ticos.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Analiza y asimila un ejemplo de este caso:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>8x<sup>2<\/sup>&nbsp;+ 6 =9y<sup>2<\/sup>&nbsp;+ 21y<br>3x<sup>2<\/sup>&nbsp;&#8211; 3 = 14y \u2212 y<\/strong><sup><strong>2<\/strong><\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplica por 3 y &#8211; 8 las ecuaciones para que elimines el t\u00e9rmino de la variable&nbsp;\u00abx<sup>2<\/sup>&nbsp;\u00ab:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.25.47.png\" alt=\"\" class=\"wp-image-1981\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"90\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.25.47.png\" alt=\"\" class=\"wp-image-1981\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Suma ambas ecuaciones: 42 = 35y<sup>2<\/sup>&nbsp;&#8211; 49y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ordena la ecuaci\u00f3n: 0 = 35y<sup>2<\/sup>&nbsp;&#8211; 49y -42<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Reduce la ecuaci\u00f3n: 5y<sup>2<\/sup>&nbsp;&#8211; 7y -6 =0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Factoriza: (5y + 3)(y &#8211; 2) =0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n para \u201cy\u201d: &nbsp;y=2, y=.3\/5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituye el valor de \u00aby = 2\u00bb en 3x<sup>2<\/sup>&nbsp;-3 = 14 y &#8211; y<sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>3x<sup>2<\/sup>&nbsp;-3 = 14 (2) -(2)<\/strong><sup><strong>2<\/strong><\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Reduce y despeja: 3x<sup>2<\/sup>&nbsp;= 28 \u2212 4 + 3 = 27<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n para \u201cx\u201d:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.26.28.png\" alt=\"\" class=\"wp-image-1982\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"165\" height=\"65\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.26.28.png\" alt=\"\" class=\"wp-image-1982\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Sustituye el valor de \u201cy =\u22123\/5\u201d en 3x<sup>2<\/sup>&nbsp;\u22123 =14y \u2212 y<sup>2<\/sup>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">3x<sup>2<\/sup>&nbsp;\u22123 =14(\u22123\/5)\u2212(\u22123\/5)<sup>2<\/sup>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Reduce y despeja: 3x<sup>2&nbsp;<\/sup>=\u221242\/5\u22129\/25 + 3 =\u2212294<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n para \u201cx\u201d &nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.27.00.png\" alt=\"\" class=\"wp-image-1983\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"76\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.27.00.png\" alt=\"\" class=\"wp-image-1983\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: (3,2),(\u22123,2)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Caso 2: Sistema con dos ecuaciones de la forma ax<sup>2<\/sup> + by<sup>2<\/sup>&nbsp;+ cx + dy + e = 0 y mx.y = n &nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En estos sistemas, es conveniente que utilices el m\u00e9todo de sustituci\u00f3n, como se muestra enseguida:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Analiza y asimila un ejemplo de este caso:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.27.32.png\" alt=\"\" class=\"wp-image-1984\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"149\" height=\"95\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.27.32.png\" alt=\"\" class=\"wp-image-1984\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">De la ecuaci\u00f3n xy = \u221a5 despeja la variable \u00aby\u00bb:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.00.png\" alt=\"\" class=\"wp-image-1985\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"99\" height=\"88\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.00.png\" alt=\"\" class=\"wp-image-1985\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Sustituye:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.24.png\" alt=\"\" class=\"wp-image-1986\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"108\" height=\"112\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.24.png\" alt=\"\" class=\"wp-image-1986\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">en la ecuaci\u00f3n x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;= 2<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.46.png\" alt=\"\" class=\"wp-image-1987\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"192\" height=\"110\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.28.46.png\" alt=\"\" class=\"wp-image-1987\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Desarrolla operaciones:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.06.png\" alt=\"\" class=\"wp-image-1988\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"82\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.06.png\" alt=\"\" class=\"wp-image-1988\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Multiplica por el denominador com\u00fan: x<sup>4<\/sup>&nbsp;+ 5 = 2x<sup>2<\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ordena la ecuaci\u00f3n: x<sup>4<\/sup>&nbsp;\u2212 2x<sup>2<\/sup>&nbsp;+ 5 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Aplica la f\u00f3rmula general:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.37.png\" alt=\"\" class=\"wp-image-1989\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"559\" height=\"281\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.37.png\" alt=\"\" class=\"wp-image-1989\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.37.png 559w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.29.37-300x151.png 300w\" sizes=\"auto, (max-width: 559px) 100vw, 559px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">*Si graficas en GeoGebra, observar\u00e1s que no hay intersecci\u00f3n porque la soluci\u00f3n es compleja*<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Caso 3: Sistema con dos ecuaciones de la forma ax<sup>2<\/sup>&nbsp;+axy+cy<sup>2<\/sup>&nbsp;=k<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En sistemas que contienen ecuaciones de la forma ax<sup>2<\/sup>&nbsp;+axy+cy<sup>2<\/sup>&nbsp;=k , es conveniente que utilices el m\u00e9todo de suma o resta para que elimines el t\u00e9rmino que no lleva variables. A continuaci\u00f3n, se muestra un ejemplo de este caso:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Analiza y asimila los siguientes ejemplos:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.30.10.png\" alt=\"\" class=\"wp-image-1990\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"170\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-23-a-las-16.30.10.png\" alt=\"\" class=\"wp-image-1990\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Suma las ecuaciones: \u22123x<sup>2<\/sup>&nbsp;\u2212 xy + 2y<sup>2<\/sup>&nbsp;= 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Factoriza: (3x-2y)(-x-y) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n de una variable en t\u00e9rminos de la otra x = 2y \/3, x =\u2212y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituye x = 2y \/ 3 en una de las ecuaciones originales:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/1-11.png\" alt=\"\" class=\"wp-image-2007\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"385\" height=\"199\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/1-11.png\" alt=\"\" class=\"wp-image-2007\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/1-11.png 385w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/1-11-300x155.png 300w\" sizes=\"auto, (max-width: 385px) 100vw, 385px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Multiplica por el denominador com\u00fan: 44y<sup>2<\/sup>&nbsp;+ 6y<sup>2&nbsp;<\/sup>\u2212 45y<sup>2<\/sup>&nbsp;= 45<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suma los t\u00e9rminos semejantes: 5y<sup>2<\/sup>&nbsp;= 45<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verifica la soluci\u00f3n: y =\u00b1 3&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x = 2(\u00b1 3) \/3 =\u00b12<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituye x=\u2212y en una de las ecuaciones originales: 11(\u2212 y)<sup>2<\/sup>&nbsp;+(\u2212 y)y \u22125y<sup>2<\/sup>&nbsp;= 5<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>11y<sup>2<\/sup>&nbsp;\u2212 y<sup>2&nbsp;<\/sup>\u22125y<sup>2<\/sup>&nbsp;= 5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suma los t\u00e9rminos semejantes: 5y<sup>2<\/sup>&nbsp;= 5.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: y =\u00b11, x = \u2212(\u00b11) = \u00b11<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: (3,2),(\u22123,\u22122),(1,\u22121),(\u22121,1)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo: Resuelve por el m\u00e9todo de suma o resta el sistema de ecuaciones:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>x<sup>2<\/sup>&nbsp;\u2212 2xy \u2212 y<sup>2<\/sup>&nbsp;= 41,<br>x<sup>2<\/sup>&nbsp;+ 3xy \u2212 y<sup>2<\/sup>&nbsp;=131<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Mediante la propiedad aditiva restan la primera y segunda ecuaci\u00f3n, de manera que elimines las variables &nbsp;x<sup>2<\/sup>&nbsp;y &nbsp;y<sup>2<\/sup>&nbsp;.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/2-10.png\" alt=\"\" class=\"wp-image-2009\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"220\" height=\"153\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/2-10.png\" alt=\"\" class=\"wp-image-2009\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Luego, aplica la propiedad multiplicativa para despejar una de las variables: y =18 \/ x<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Despu\u00e9s, aplica la propiedad de sustituci\u00f3n y sustituye la variable en cualquiera de las ecuaciones:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/3-8.png\" alt=\"\" class=\"wp-image-2021\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"665\" height=\"75\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/3-8.png\" alt=\"\" class=\"wp-image-2021\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/3-8.png 665w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/3-8-300x34.png 300w\" sizes=\"auto, (max-width: 665px) 100vw, 665px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Aplica la propiedad multiplicativa: &nbsp;x<sup>4<\/sup>&nbsp;\u221236x<sup>2<\/sup>&nbsp;\u2212324 = 41x<sup>2<\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Por la propiedad aditiva: &nbsp;x<sup>4<\/sup>&nbsp;\u2212 77x<sup>2<\/sup>&nbsp;\u2212324 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Mediante factorizaci\u00f3n: (x<sup>2<\/sup>&nbsp;\u221281)(x<sup>2<\/sup>&nbsp;+ =4) &nbsp;0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n real: &nbsp; x=\u00b19 , {(9,2), (-9,- 2)}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como siguiente actividad, te invito a analizar los siguientes videos en donde se explica a detalle c\u00f3mo resolver sistemas de ecuaciones cuadr\u00e1tica \u2013 cuadr\u00e1tica.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"\u00bfSabes resolver sistemas de ecuaciones de segundo grado?\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/71maaFFtZ2o?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/es.khanacademy.org\/math\/eb-1-semestre-bachillerato\/eb-quadratic-equations-and-functions\/eb-systems-of-quadratic-equations\/v\/non-linear-systems-of-equations-2\" target=\"_blank\" rel=\"noreferrer noopener\">Sistemas cuadr\u00e1ticos: soluci\u00f3n algebraica<\/a><\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"1-3-planteamiento-y-solucion-de-problemas\">1.3 Planteamiento y soluci\u00f3n de problemas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">En este tema se presenta la aplicaci\u00f3n de las ecuaciones, la cual comprende la soluci\u00f3n de problemas que describen cantidades conocidas, cantidades desconocidas y las relaciones entre ellas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En la uni\u00f3n de la teor\u00eda con la pr\u00e1ctica surgen problemas relacionados con n\u00fameros, porcentajes, edades, velocidades, figuras geom\u00e9tricas, etc. Para la soluci\u00f3n de estos problemas es necesario:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Identificar la(s) variable(s) o inc\u00f3gnita(s) del problema y representarla(s) con una variable.<\/li>\n\n\n\n<li>Escribir un modelo matem\u00e1tico (ecuaciones) que represente las relaciones entre la variable y las cantidades conocidas.<\/li>\n\n\n\n<li>Resolver el modelo matem\u00e1tico (soluci\u00f3n de ecuaciones o sistema de ecuaciones) y comprobar la soluci\u00f3n.&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">A continuaci\u00f3n, se presenta una variable, la relaci\u00f3n entre la variable y cantidades conocidas, as\u00ed como el modelo matem\u00e1tico que muestra esa relaci\u00f3n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sea &nbsp;\u00abx\u00bb &nbsp;la edad actual de Juan.<br>Su edad dentro de 10 a\u00f1os se expresa como: x+10.<br>Dentro de 10 a\u00f1os su edad ser\u00e1 el triple de su edad actual: x+10=3x.<br>Sea \u00aba\u00bb la edad actual de Dora y \u00abb\u00bb la edad actual de Diego.<br>El d\u00eda de hoy la edad de Dora es cinco sextos de la edad de Diego:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/4-7.png\" alt=\"\" class=\"wp-image-2026\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"98\" height=\"100\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/4-7.png\" alt=\"\" class=\"wp-image-2026\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Dentro de 5 a\u00f1os la edad de Dora ser\u00e1 cinco tercios de la edad actual de Diego:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/5-6.png\" alt=\"\" class=\"wp-image-2027\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"125\" height=\"81\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/5-6.png\" alt=\"\" class=\"wp-image-2027\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Sea &nbsp;\u00aba\u00bb &nbsp;el ancho y \u00abb\u00bb &nbsp;el largo en metros de un rect\u00e1ngulo.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El per\u00edmetro es 540 metros: a+a+b+b= 540 m &nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El largo es 30 metros mayor que el doble del ancho: b = 30 m + 2a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sea &nbsp;\u00aba\u00bb &nbsp;el ancho y \u00abb\u00bb &nbsp;el largo en metros de un rect\u00e1ngulo.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El ancho es dos tercios del largo:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/6-6.png\" alt=\"\" class=\"wp-image-2028\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"90\" height=\"82\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/6-6.png\" alt=\"\" class=\"wp-image-2028\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Si las dimensiones aumentan 2 metros el \u00e1rea aumenta 64 m<sup>2<\/sup>: (a+ 2)(b+ 2) = (a)(b) + 64 m<sup>2<\/sup><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sean &nbsp;\u00aba\u00bb &nbsp;y \u00abb\u00bb &nbsp;dos n\u00fameros.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La diferencia entre ambos es 24: a\u2212b=24<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Al dividir el mayor entre el menor el cociente es 2 y el residuo es 9:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/7-5.png\" alt=\"\" class=\"wp-image-2030\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"122\" height=\"71\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/7-5.png\" alt=\"\" class=\"wp-image-2030\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Sean &nbsp;\u00aba\u00bb &nbsp;el d\u00edgito de las decenas y \u00abb\u00bb &nbsp;el d\u00edgito de las unidades.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El d\u00edgito de las decenas supera en 3 al d\u00edgito de las unidades: a=b+3<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El n\u00famero supera en 8 al s\u00e9xtuplo de la suma de los d\u00edgitos: 10a +b = 6(a +b) +8<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como ejemplo considera los siguientes problemas:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La diferencia de dos n\u00fameros naturales es 4 y la suma de sus rec\u00edprocos es 4\/15. Escribe el modelo matem\u00e1tico y la soluci\u00f3n que determina dichos n\u00fameros naturales.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sean &nbsp;\u00abx\u00bb e \u00aby\u00bb dos n\u00fameros naturales y sus respectivos rec\u00edprocos.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/8-5.png\" alt=\"\" class=\"wp-image-2031\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"141\" height=\"81\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/8-5.png\" alt=\"\" class=\"wp-image-2031\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">El modelo matem\u00e1tico es: <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/9-5.png\" alt=\"\" class=\"wp-image-2032\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"140\" height=\"116\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/9-5.png\" alt=\"\" class=\"wp-image-2032\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">La soluci\u00f3n del modelo matem\u00e1tico es por medio del m\u00e9todo de sustituci\u00f3n, tal y como se muestra a continuaci\u00f3n:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De la ecuaci\u00f3n &nbsp;x \u2212 y = 4 &nbsp;se despeja una variable: &nbsp;x = 4 + y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Luego, &nbsp;x = 4 + y se sustituye en la otra ecuaci\u00f3n:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/10-4.png\" alt=\"\" class=\"wp-image-2033\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"166\" height=\"82\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/10-4.png\" alt=\"\" class=\"wp-image-2033\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se multiplica por el denominador com\u00fan: 1 (15y)+1 (15)(4+y) 4(y)(y + 4)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se desarrollan las operaciones: &nbsp;15y + 60 + 15y = 4y<sup>2<\/sup>&nbsp;+16y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se reducen y se ordenan los t\u00e9rminos: &nbsp; 0 = 4y<sup>2<\/sup>&nbsp;\u221214y \u2212 60<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Los coeficientes admiten mitad: &nbsp;2y<sup>2 &nbsp;<\/sup>-7y &#8211; 30 = 0&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Por factorizaci\u00f3n: (2y+ 5)(y\u2212 6) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: &nbsp;y = 6<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo &nbsp;y = 6 en una de las ecuaciones: &nbsp; x = 4 + 6 = 10<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: x =10, y = 6<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Un n\u00famero entero de dos d\u00edgitos es 12 unidades mayor que el qu\u00edntuplo de la suma de sus d\u00edgitos. Adem\u00e1s, el cuadrado de la suma de sus d\u00edgitos es 39 unidades menos que el triple del siguiente n\u00famero entero. Determina los d\u00edgitos del n\u00famero entero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sean &nbsp;\u00abd\u00bb &nbsp;el d\u00edgito de las decenas y \u00abu\u00bb el d\u00edgito de las unidades:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">10d u = 12 + 5 (d + u)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(d +u)<sup>2<\/sup>&nbsp;=3(10d + u+1) &#8211; 39<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se desarrollan las operaciones de la ecuaci\u00f3n: &nbsp;10d + u = 12 + 5(d + u)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">10d + u = 12 + 5d + 5u<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se despeja una variable:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/11-3.png\" alt=\"\" class=\"wp-image-2034\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"132\" height=\"79\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/11-3.png\" alt=\"\" class=\"wp-image-2034\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se sustituye:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/11-4.png\" alt=\"\" class=\"wp-image-2035\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"132\" height=\"79\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/11-4.png\" alt=\"\" class=\"wp-image-2035\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">En la ecuaci\u00f3n (d + u)<sup>2<\/sup>&nbsp;= 3( 10d + u +1) -39&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/12-3.png\" alt=\"\" class=\"wp-image-2036\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"461\" height=\"99\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/12-3.png\" alt=\"\" class=\"wp-image-2036\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/12-3.png 461w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/12-3-300x64.png 300w\" sizes=\"auto, (max-width: 461px) 100vw, 461px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se desarrollan operaciones:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/13-4.png\" alt=\"\" class=\"wp-image-2037\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"357\" height=\"256\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/13-4.png\" alt=\"\" class=\"wp-image-2037\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/13-4.png 357w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/13-4-300x215.png 300w\" sizes=\"auto, (max-width: 357px) 100vw, 357px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se reducen y ordenan los t\u00e9rminos: 81u<sup>2<\/sup>&nbsp;\u2212 459u \u2212 756 = 0<br>Los coeficientes son divisibles entre 27: 3u<sup>2<\/sup>&nbsp;\u221217u \u2212 28 = 0<br>Por factorizaci\u00f3n:(3u+ 4)(u\u2212 7) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: &nbsp;u =7<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo &nbsp;u =7 en la ecuaci\u00f3n:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/14-3.png\" alt=\"\" class=\"wp-image-2038\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"271\" height=\"83\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/14-3.png\" alt=\"\" class=\"wp-image-2038\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: d =8 decenas, &nbsp;u =7 unidades<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Una persona cobr\u00f3 $1920 por hacer un trabajo. Pero, al realizar el trabajo necesito 4 horas m\u00e1s de lo planeado, y por consiguiente gan\u00f3 $ 24 menos por hora. \u00bfEn cu\u00e1nto tiempo plane\u00f3 realizar el trabajo?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sean &nbsp;\u00abx\u00bb &nbsp;el n\u00famero de horas de trabajo planeadas e \u00aby\u00bb el pago por hora.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x.y =1920<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(x+ 4)(y\u2212 24) =1920<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De la ecuaci\u00f3n x y. =1920, se despeja una variable:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/15-1.png\" alt=\"\" class=\"wp-image-2039\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"122\" height=\"76\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/15-1.png\" alt=\"\" class=\"wp-image-2039\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se sustituye:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/15-2.png\" alt=\"\" class=\"wp-image-2040\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"122\" height=\"76\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/15-2.png\" alt=\"\" class=\"wp-image-2040\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">en la ecuaci\u00f3n&nbsp;(x+ 4)(y\u2212 24) =1920.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/16-1.png\" alt=\"\" class=\"wp-image-2041\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"294\" height=\"92\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/16-1.png\" alt=\"\" class=\"wp-image-2041\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se multiplica por el denominador com\u00fan: (1920 + 4y)(y \u2212 24) =1920y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se desarrollan las operaciones:&nbsp;1920y \u2212 46080 + 4y<sup>2<\/sup>&nbsp;\u221296y =1920y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se ordenan los t\u00e9rminos de la ecuaci\u00f3n: &nbsp; 4y<sup>2<\/sup>&nbsp;\u221296y \u2212 46080 = 0&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se divide entre 4: &nbsp;y<sup>2<\/sup>&nbsp;\u2212 24y \u221211520 = 0&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Factorizando:&nbsp; (y\u2212120)(y+ 96) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: y = $120 por hora<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo y = 120 en la ecuaci\u00f3n<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/17-1.png\" alt=\"\" class=\"wp-image-2042\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"303\" height=\"90\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/17-1.png\" alt=\"\" class=\"wp-image-2042\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/17-1.png 303w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/17-1-300x89.png 300w\" sizes=\"auto, (max-width: 303px) 100vw, 303px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: x =16 horas<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Un ciclista pedalea durante 30 Km a cierta velocidad. Despu\u00e9s aumenta su velocidad en 20 Km por hora durante los siguientes 30 Km. Si el recorrido total lo realiz\u00f3 en 1 \u00bc horas. \u00bfA qu\u00e9 velocidad recorri\u00f3 los primeros 30 Km?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sea &nbsp;\u00abv\u00bb &nbsp;la velocidad en los primeros 30 Km<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como la velocidad es distancia entre tiempo:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/18.png\" alt=\"\" class=\"wp-image-2043\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"81\" height=\"80\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/18.png\" alt=\"\" class=\"wp-image-2043\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">entonces<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/19.png\" alt=\"\" class=\"wp-image-2044\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"68\" height=\"88\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/19.png\" alt=\"\" class=\"wp-image-2044\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Tiempo que emplea en los primeros 30 Km:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/20.png\" alt=\"\" class=\"wp-image-2045\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"76\" height=\"78\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/20.png\" alt=\"\" class=\"wp-image-2045\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Tiempo que emplea en los siguientes 30 Km:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/21.png\" alt=\"\" class=\"wp-image-2046\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"114\" height=\"80\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/21.png\" alt=\"\" class=\"wp-image-2046\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Tiempo total del recorrido:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/22.png\" alt=\"\" class=\"wp-image-2047\" style=\"width:208px;height:86px\" width=\"208\" height=\"86\" \/><noscript><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/22.png\" alt=\"\" class=\"wp-image-2047\" style=\"width:208px;height:86px\" width=\"208\" height=\"86\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Se aplica la propiedad multiplicativa: &nbsp;1.25(v)(v + 20) = 30(v+ 20)+ 30(v)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se desarrollan las operaciones: &nbsp;1.25v<sup>2<\/sup>&nbsp;+ 25v = 30v + 600 + 30v<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Se agrupan t\u00e9rminos y se ordenan: 1.25v<sup>2<\/sup>&nbsp;\u221235v \u2212 600 = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Por factorizaci\u00f3n: (1.25v+15)(v\u2212 40) = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n: v=40 Km\/h<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"conclusion\">Conclusi\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">En conclusi\u00f3n, las soluciones de sistemas de ecuaciones cuadr\u00e1ticas, son los puntos de intersecci\u00f3n de sus gr\u00e1ficas o los lugares donde las ecuaciones tienen los mismos valores de las variables.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Recuerda, como es frecuente que, en alguno de los pasos de la resoluci\u00f3n, se haya tenido que elevar al cuadrado alguna de las inc\u00f3gnitas, se habr\u00e1n introducido as\u00ed soluciones \u00abextra\u00f1as\u00bb. Es imprescindible comprobar todas y cada una de las parejas de ra\u00edces o soluciones obtenidas para las inc\u00f3gnitas en las ecuaciones originales del sistema. Siempre habr\u00e1 que desechar alguno de estos pares, si no cumple la igualdad.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00a1Has concluido la \u00faltima clase del curso! Ha sido un gozo compartir contigo este trayecto formativo. Deseo que el curso haya cumplido con tus expectativas y te encuentres satisfecho con los temas abordados, as\u00ed como con tu desempe\u00f1o y compromiso. No olvides realizar la tarea asignada para la plena conclusi\u00f3n del curso. Espero encontrarte nuevamente, \u00a1hasta pronto!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"fuentes-de-informacion\">Fuentes de informaci\u00f3n<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Aguilar, A. (2009). Matem\u00e1ticas Simplificadas. (2a ed.). M\u00e9xico: Pearson. Cap. 12, pp. 521-524.<\/li>\n\n\n\n<li>S\u00e1nchez, F., Villanueva, R., &amp; Lugo, J. (2012). \u00c1lgebra II. M\u00e9xico: Ediciones Anglo. Cap. 3, pp. 152-156.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introducci\u00f3n Te doy la bienvenida a tu \u00faltima clase del curso de \u00c1lgebra II titulada \u201cM\u00e9todos de soluci\u00f3n de sistemas de ecuaciones cuadr\u00e1ticas con dos variables\u201d. Recuerda como lo vimos en la clase 7 y 8, un sistema de ecuaciones es un conjunto de dos o m\u00e1s ecuaciones que comparten dos o m\u00e1s inc\u00f3gnitas, donde &#8230; <a title=\"Clase digital 9: M\u00e9todos de soluci\u00f3n de sistemas de ecuaciones cuadr\u00e1ticas con dos variables\" class=\"read-more\" href=\"https:\/\/blogs.ugto.mx\/rea\/clase-digital-9-metodos-de-solucion-de-sistemas-de-ecuaciones-cuadraticas-con-dos-variables\/\" aria-label=\"Leer m\u00e1s sobre Clase digital 9: M\u00e9todos de soluci\u00f3n de sistemas de ecuaciones cuadr\u00e1ticas con dos variables\">Leer m\u00e1s<\/a><\/p>\n","protected":false},"author":142,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","episode_type":"","audio_file":"","podmotor_file_id":"","podmotor_episode_id":"","cover_image":"","cover_image_id":"","duration":"","filesize":"","filesize_raw":"","date_recorded":"","explicit":"","block":"","itunes_episode_number":"","itunes_title":"","itunes_season_number":"","itunes_episode_type":"","footnotes":""},"categories":[83,84],"tags":[41,85,86],"class_list":["post-1976","post","type-post","status-publish","format-standard","hentry","category-bachillerato-general","category-algebra-ii","tag-clase-digital","tag-neba04002","tag-sandra-patricia-arriaga-martinez"],"acf":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/1976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/users\/142"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/comments?post=1976"}],"version-history":[{"count":5,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/1976\/revisions"}],"predecessor-version":[{"id":29841,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/1976\/revisions\/29841"}],"wp:attachment":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/media?parent=1976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/categories?post=1976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/tags?post=1976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}