{"id":11137,"date":"2022-06-21T14:38:49","date_gmt":"2022-06-21T09:08:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11137"},"modified":"2022-06-21T14:38:50","modified_gmt":"2022-06-21T09:08:50","slug":"for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/","title":{"rendered":"For what values of a and b, the following system of equations have infinite number of solutions ?"},"content":{"rendered":"

Question<\/strong> :<\/h2>\n

(i)<\/strong>\u00a0 For what values of a and b, the following system of equations have infinite number of solutions ?<\/p>\n

2x + 3y = 7<\/p>\n

(a – b)x + (a + b)y = 3a + b – 2<\/p>\n

(ii)<\/strong>\u00a0 For what value of k will the following pair of linear equations have no solution ?<\/p>\n

3x + y = 1<\/p>\n

(2k – 1)x + (k – 1)y = 2k + 1<\/p>\n

Solution :<\/h2>\n

(i)<\/strong>\u00a0 The given linear equations can be written as<\/p>\n

2x + 3y – 7 = 0<\/p>\n

(a – b)x + (a + b)y – (3a + b – 2) = 0<\/p>\n

Above equations is of the form<\/p>\n

\\(a_1x + b_1y + c_1\\) = 0<\/p>\n

\\(a_2x + b_2y + c_2\\) = 0<\/p>\n

where,<\/p>\n

\\(a_1\\) = 2,\u00a0 \\(b_1\\) = 3,\u00a0 \\(c_1\\) = -7<\/p>\n

\\(a_2\\) = (a – b),\u00a0 \\(b_2\\) = (a + b),\u00a0 \\(c_2\\) = -(3a + b – 2)<\/p>\n

For the linear equations to have infinitely many solutions,<\/p>\n

\\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n

Here,\u00a0 \\(a_1\\over a_2\\) = \\(2\\over a – b\\),\u00a0 \\(b_1\\over b_2\\) = \\(3\\over a + b\\),\u00a0 \\(c_1\\over c_2\\) = \\(-7\\over -(3a + b – 2)\\) = \\(7\\over 3a + b – 2\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(2\\over a – b\\) = \\(3\\over a + b\\) = \\(7\\over 3a + b – 2\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(2\\over a – b\\) = \\(3\\over a + b\\)\u00a0 \u00a0and\u00a0 \\(3\\over a + b\\) = \\(7\\over 3a + b – 2\\)<\/p>\n

\\(\\implies\\)\u00a0 2a + 2b = 3a – 3b\u00a0 \u00a0and\u00a0 \u00a0 9a + 3b – 6 = 7a + 7b<\/p>\n

\\(\\implies\\)\u00a0 2a – 3a = -3b – 2b\u00a0 \u00a0 and\u00a0 \u00a0 \u00a09a – 7a = 7b – 3b + 6<\/p>\n

\\(\\implies\\)\u00a0 a = 5b\u00a0 \u00a0……(1)\u00a0 and\u00a0 a = 2b + 3\u00a0 \u00a0 ….(2)<\/p>\n

Solving equation (1) and (2), we get<\/p>\n

5b = 2b + 3 \\(\\implies\\)\u00a0 b = 1<\/p>\n

Substituting the value of b in equation (1), we get<\/p>\n

a = 5<\/p>\n

Thus,\u00a0 a = 5 and b = 1.\u00a0<\/strong><\/p>\n

(ii)<\/strong>\u00a0 The given linear equations can be written as<\/p>\n

3x + y – 1 = 0<\/p>\n

(2k – 1)x + (k – 1)y – (2k + 1) = 0<\/p>\n

Above equations is of the form<\/p>\n

\\(a_1x + b_1y + c_1\\) = 0<\/p>\n

\\(a_2x + b_2y + c_2\\) = 0<\/p>\n

where,<\/p>\n

\\(a_1\\) = 3,\u00a0 \\(b_1\\) = 1,\u00a0 \\(c_1\\) = -1<\/p>\n

\\(a_2\\) = (2k – 1),\u00a0 \\(b_2\\) = (k – 1),\u00a0 \\(c_2\\) = -(2k + 1)<\/p>\n

For the linear equations to have no solutions,<\/p>\n

\\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) \\(\\ne\\) \\(c_1\\over c_2\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(3\\over 2k -1\\) = \\(1\\over k – 1\\)\u00a0 \u00a0and\u00a0 \\(1\\over k – 1\\) \\(\\ne\\)\u00a0 \\(-1\\over -(2k + 1\\)\u00a0 \u00a0 and\u00a0 \\(3\\over 2k -1\\) \\(\\ne\\)\u00a0 \\(1\\over (2k + 1\\)<\/p>\n

\\(\\implies\\)\u00a0 3k – 3 = 2k – 1\u00a0 and\u00a0 \u00a02k + 1 \\(\\ne\\)\u00a0 k – 1\u00a0 and\u00a0 6k + 3 \\(\\ne\\)\u00a0 2k – 1<\/p>\n

\\(\\implies\\)\u00a0 k = 2\u00a0 \u00a0and\u00a0 \u00a0k\\(\\ne\\)\u00a0 -2\u00a0 \u00a0and\u00a0 k \\(\\ne\\) -1<\/p>\n

Hence, the given linear equations has no solutions for k = 2\u00a0 \u00a0and\u00a0 \u00a0k\\(\\ne\\)\u00a0 -2\u00a0 \u00a0and\u00a0 k \\(\\ne\\) -1<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Question : (i)\u00a0 For what values of a and b, the following system of equations have infinite number of solutions ? 2x + 3y = 7 (a – b)x + (a + b)y = 3a + b – 2 (ii)\u00a0 For what value of k will the following pair of linear equations have no solution …<\/p>\n

For what values of a and b, the following system of equations have infinite number of solutions ?<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[911,43],"tags":[],"yoast_head":"\nFor what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : (i)\u00a0 For what values of a and b, the following system of equations have infinite number of solutions ? 2x + 3y = 7 (a – b)x + (a + b)y = 3a + b – 2 (ii)\u00a0 For what value of k will the following pair of linear equations have no solution … For what values of a and b, the following system of equations have infinite number of solutions ? Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-06-21T09:08:49+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-06-21T09:08:50+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"For what values of a and b, the following system of equations have infinite number of solutions ?\",\"datePublished\":\"2022-06-21T09:08:49+00:00\",\"dateModified\":\"2022-06-21T09:08:50+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\"},\"wordCount\":393,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Linear Equation in Two Variables Questions\",\"Maths Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\",\"url\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\",\"name\":\"For what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2022-06-21T09:08:49+00:00\",\"dateModified\":\"2022-06-21T09:08:50+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"For what values of a and b, the following system of equations have infinite number of solutions ?\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"For what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize","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:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/","og_locale":"en_US","og_type":"article","og_title":"For what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize","og_description":"Question : (i)\u00a0 For what values of a and b, the following system of equations have infinite number of solutions ? 2x + 3y = 7 (a – b)x + (a + b)y = 3a + b – 2 (ii)\u00a0 For what value of k will the following pair of linear equations have no solution … For what values of a and b, the following system of equations have infinite number of solutions ? Read More »","og_url":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/","og_site_name":"Mathemerize","article_published_time":"2022-06-21T09:08:49+00:00","article_modified_time":"2022-06-21T09:08:50+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"For what values of a and b, the following system of equations have infinite number of solutions ?","datePublished":"2022-06-21T09:08:49+00:00","dateModified":"2022-06-21T09:08:50+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/"},"wordCount":393,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Linear Equation in Two Variables Questions","Maths Questions"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/","url":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/","name":"For what values of a and b, the following system of equations have infinite number of solutions ? - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2022-06-21T09:08:49+00:00","dateModified":"2022-06-21T09:08:50+00:00","breadcrumb":{"@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/for-what-values-of-a-and-b-the-following-system-of-equations-have-infinite-number-of-solutions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"For what values of a and b, the following system of equations have infinite number of solutions ?"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11137"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=11137"}],"version-history":[{"count":2,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11137\/revisions"}],"predecessor-version":[{"id":11140,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11137\/revisions\/11140"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=11137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=11137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=11137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}