{"id":225178,"date":"2025-06-03T21:46:01","date_gmt":"2025-06-03T21:46:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225178"},"modified":"2025-06-03T21:46:03","modified_gmt":"2025-06-03T21:46:03","slug":"how-many-solutions-exist-for-the-system-of-equations-below","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/03\/how-many-solutions-exist-for-the-system-of-equations-below\/","title":{"rendered":"How many solutions exist for the system of equations below"},"content":{"rendered":"\n

How many solutions exist for the system of equations below?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

We are given the system of equations:<\/p>\n\n\n\n

    \n
  1. 3x+y=183x + y = 18<\/li>\n\n\n\n
  2. 3x+y=163x + y = 16<\/li>\n<\/ol>\n\n\n\n

    To determine how many solutions exist, let\u2019s analyze these two equations.<\/p>\n\n\n\n

    Both equations have the same left-hand side<\/strong>: 3x+y3x + y, but the right-hand sides are different<\/strong> (18 and 16). This suggests that the lines represented by these equations are parallel<\/strong> but have different y-intercepts<\/strong>.<\/p>\n\n\n\n

    Step-by-step Explanation:<\/h3>\n\n\n\n

    Let’s subtract the second equation from the first: (3x+y)\u2212(3x+y)=18\u221216(3x + y) – (3x + y) = 18 – 16 0=20 = 2<\/p>\n\n\n\n

    This is a contradiction<\/strong>. The result, 0=20 = 2, is false and shows that the system of equations has no solution<\/strong>.<\/p>\n\n\n\n

    Why?<\/h3>\n\n\n\n

    Graphically, both equations represent straight lines. Since their left-hand sides (slopes) are the same, they are parallel lines<\/strong>. However, because their right-hand sides (y-intercepts) differ, they never intersect<\/strong>.<\/p>\n\n\n\n

    In a system of linear equations:<\/p>\n\n\n\n

      \n
    • If the lines intersect at exactly one point \u2192 One solution<\/strong><\/li>\n\n\n\n
    • If the lines are the same (overlap completely) \u2192 Infinitely many solutions<\/strong><\/li>\n\n\n\n
    • If the lines are parallel and distinct \u2192 No solution<\/strong><\/li>\n<\/ul>\n\n\n\n

      This system falls into the third case.<\/p>\n\n\n\n

      Final Answer:<\/h3>\n\n\n\n

      None<\/strong><\/p>\n\n\n\n

      This means the system is inconsistent<\/strong>, and the two equations contradict each other. There is no pair of values<\/strong> for xx and yy that will satisfy both equations simultaneously.<\/p>\n\n\n\n

      Summary:<\/h3>\n\n\n\n

      The equations 3x+y=183x + y = 18 and 3x+y=163x + y = 16 describe two parallel but different lines<\/strong>. Because they never cross, there is no point (x, y)<\/strong> that solves both equations at once.<\/p>\n\n\n\n

      \u2705 Correct answer: none<\/strong><\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      How many solutions exist for the system of equations below? The Correct Answer and Explanation is: We are given the system of equations: To determine how many solutions exist, let\u2019s analyze these two equations. Both equations have the same left-hand side: 3x+y3x + y, but the right-hand sides are different (18 and 16). This suggests […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-225178","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=225178"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225178\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}