How many solutions does this system of equations have?
none
exactly two
infinitely many
exactly one
Graph of a system of linear equations. Equation 1 is 2x plus y equals 1.. Equation 2 is 4x plus 2y equals 2. The graphs are the same line.<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To determine how many solutions the given system of equations has, let’s first analyze the equations provided:<\/p>\n\n\n\n Next, we can simplify the second equation. Dividing every term in the second equation by 2 yields:<\/p>\n\n\n\n [ Now, we can see that both equations are identical:<\/p>\n\n\n\n Since both equations represent the same line on a graph, this means that every point on this line is a solution to both equations. In geometric terms, when two linear equations represent the same line, they intersect at every point along that line.<\/p>\n\n\n\n In terms of solutions, this implies that there are infinitely many solutions to the system of equations. Each point ((x, y)) on the line (2x + y = 1) is a valid solution, as substituting any of these points into either equation will satisfy both equations.<\/p>\n\n\n\n To visualize this, if you were to graph both equations, you would see a single line where both equations lie. Therefore, the correct answer to the question of how many solutions this system of equations has is infinitely many<\/strong>.<\/p>\n\n\n\n In conclusion, because both equations represent the same line, they share all their points as solutions, leading to an infinite number of solutions. This characteristic is typical of dependent systems, where the equations do not merely intersect at one point but overlap entirely, reflecting that they are not independent equations.<\/p>\n","protected":false},"excerpt":{"rendered":" How many solutions does this system of equations have?noneexactly twoinfinitely manyexactly oneGraph of a system of linear equations. Equation 1 is 2x plus y equals 1.. Equation 2 is 4x plus 2y equals 2. The graphs are the same line. The Correct Answer and Explanation is: To determine how many solutions the given system of […]<\/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-161507","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/161507","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=161507"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/161507\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=161507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=161507"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=161507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\\frac{4x}{2} + \\frac{2y}{2} = \\frac{2}{2} \\quad \\Rightarrow \\quad 2x + y = 1
]<\/p>\n\n\n\n\n