{"id":108,"date":"2022-04-13T10:01:05","date_gmt":"2022-04-13T10:01:05","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=108"},"modified":"2024-12-20T06:47:14","modified_gmt":"2024-12-20T06:47:14","slug":"e-5-find-the-number-of-solutions-to-a-pair-of-equations","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/e-5-find-the-number-of-solutions-to-a-pair-of-equations\/","title":{"rendered":"E.5 Find the number of solutions to a pair of equations"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color has-huge-font-size\" style=\"color:#00056d;text-transform:uppercase\"><strong>Find the number of solutions to a pair of equations<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-4dde4300b7763c9a2a860fe18546dfe2\" style=\"color:#74008b;text-transform:uppercase\">key notes:<\/p>\n\n\n\n<div class=\"wp-block-group has-background-background-color has-background has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>\ud83d\udca1A system of two equations can be classified as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the slopes are the same but the&nbsp;<em>y<\/em>-intercepts&nbsp;are different, the system has no solution.<\/li>\n\n\n\n<li>If the slopes are different, the system has one solution.<\/li>\n\n\n\n<li>If the slopes are the same and the&nbsp;<em>y<\/em>-intercepts&nbsp;are the same, the system has infinitely many solutions.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading has-large-font-size\">Three Types of Solutions of a System of Linear Equations:<\/h4>\n\n\n\n<p class=\"has-large-font-size\">Consider the pair of linear equations in two variables x and y.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub>&nbsp;= 0<\/li>\n\n\n\n<li class=\"has-large-font-size\">a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub>&nbsp;= 0<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">Here a<sub>1<\/sub>, b<sub>1<\/sub>, c<sub>1<\/sub>, a<sub>2<\/sub>, b<sub>2<\/sub>, c<sub>2&nbsp;<\/sub>are all real numbers.<\/p>\n\n\n\n<p class=\"has-large-font-size\">Note that, a<sub>1<\/sub><sup>2<\/sup>&nbsp;+ b<sub>1<\/sub><sup>2<\/sup>&nbsp;\u2260 0, a<sub>2<\/sub><sup>2<\/sup>&nbsp;+ b<sub>2<\/sub><sup>2<\/sup>&nbsp;\u2260 0<\/p>\n\n\n\n<p class=\"has-large-font-size\">1. If (a<sub>1<\/sub>\/a<sub>2<\/sub>) \u2260 (b<sub>1<\/sub>\/b<sub>2<\/sub>), then there will be a&nbsp;<strong>unique solution<\/strong>. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations.<\/p>\n\n\n\n<p class=\"has-large-font-size\">2. If (a<sub>1<\/sub>\/a<sub>2<\/sub>) = (b<sub>1<\/sub>\/b<sub>2<\/sub>) = (c<sub>1<\/sub>\/c<sub>2<\/sub>), then there will be&nbsp;<strong>infinitely many solutions<\/strong>. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables<\/p>\n\n\n\n<p class=\"has-large-font-size\">3. If (a<sub>1<\/sub>\/a<sub>2<\/sub>) = (b<sub>1<\/sub>\/b<sub>2<\/sub>) \u2260 (c<sub>1<\/sub>\/c<sub>2<\/sub>), then there will be&nbsp;<strong>no<\/strong>&nbsp;<strong>solution<\/strong>. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations.<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/h4>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f3e593\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-3997c624e20b0d57c9cba7323337ec28\" style=\"color:#b00012\">How many solutions does the following system have?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>y = -2x \u2013 4<\/li>\n\n\n\n<li>y = 3x + 3<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Given y = -2x \u2013 4<\/p>\n\n\n\n<p>y = 3x + 3<\/p>\n\n\n\n<p>Rewriting to the general form<\/p>\n\n\n\n<p>-2x \u2013 y \u2013 4 = 0<\/p>\n\n\n\n<p>3x \u2013 y + 3 = 0<\/p>\n\n\n\n<p>Comparing the coefficients,<\/p>\n\n\n\n<p>(a<sub>1<\/sub>\/a<sub>2<\/sub>) = -\u2154<\/p>\n\n\n\n<p>(b<sub>1<\/sub>\/b<sub>2<\/sub>) = -1\/-1 = 1<\/p>\n\n\n\n<p>(a<sub>1<\/sub>\/a<sub>2<\/sub>) \u2260 (b<sub>1<\/sub>\/b<sub>2<\/sub>)<\/p>\n\n\n\n<p>Hence, this system of equations will have only one solution.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f39d9d\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color wp-elements-ca9558a621211907e59659c2497f8a1e\" style=\"color:#b00012\">How many solutions do the simultaneous equations below have?<\/p>\n\n\n\n<p>y = 5x + 2<\/p>\n\n\n\n<p>y=7\/3x+9\/2<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>no solution<\/li>\n\n\n\n<li>one solution<\/li>\n\n\n\n<li>infinitely many solutions<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p>The first equation is y = 5x + 2, so the slope is 5 and the y-intercept is 2.<br>The second equation is y =7\/3x +9\/2, so the slope is 7\/3 and the y-intercept is 9\/2.<\/p>\n\n\n\n<p>The slopes are different, so the lines intersect at one point. The system has one<br><\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#8feeac\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color wp-elements-ca9558a621211907e59659c2497f8a1e\" style=\"color:#b00012\">How many solutions do the simultaneous equations below have?<\/p>\n\n\n\n<p>y = 6x \u2212 2<br>y=6x\u22122\/7<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>no solution<\/li>\n\n\n\n<li>one solution<\/li>\n\n\n\n<li>infinitely many solutions<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p>The first equation is y = 6x \u2212 2, so the slope is 6 and the y-intercept is -2.<\/p>\n\n\n\n<p>The second equation is y = 6x \u22122\/7, so the slope is 6 and the y-intercept is -2\/7.<\/p>\n\n\n\n<p>The slopes are the same but the y-intercepts are different, so the lines are<br>parallel and never intersect. The system has no solution.<\/p>\n<\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading has-text-color has-link-color wp-elements-8eef9b304c5fe333c52d54e442bd198d\" id=\"yui_3_18_1_1_1670028809318_7623\" style=\"color:#d90000\">Let&#8217;s practice:<\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/84106\/926\/334\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-24.png\" alt=\"\" class=\"wp-image-6673\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-24.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-24-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-24-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/37823\/370\/513\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-36.png\" alt=\"\" class=\"wp-image-6670\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-36.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-36-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-36-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Find the number of solutions to a pair of equations key notes: \ud83d\udca1A system of two equations can be classified as follows: Three Types of Solutions of a System of Linear Equations: Consider the pair of linear equations in two variables x and y. Here a1, b1, c1, a2, b2, c2&nbsp;are all real numbers. Note<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/e-5-find-the-number-of-solutions-to-a-pair-of-equations\/\">Continue reading <span class=\"screen-reader-text\">&#8220;E.5 Find the number of solutions to a pair of equations&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-108","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=108"}],"version-history":[{"count":20,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108\/revisions"}],"predecessor-version":[{"id":15441,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108\/revisions\/15441"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}