{"id":6009,"date":"2022-01-28T17:07:52","date_gmt":"2022-01-28T17:07:52","guid":{"rendered":"https:\/\/oryxlearning.com\/company\/?page_id=6009"},"modified":"2022-01-28T17:07:52","modified_gmt":"2022-01-28T17:07:52","slug":"how-to-solve-linear-systems-of-equations-by-substitution","status":"publish","type":"page","link":"https:\/\/oryxlearning.com\/learn\/how-to-solve-linear-systems-of-equations-by-substitution\/","title":{"rendered":"How to Solve Linear Systems of Equations by Substitution"},"content":{"rendered":"[vc_section full_width=”stretch_row” content_placement=”middle” css=”.vc_custom_1633364343825{background-image: url(https:\/\/oryxlearning.com\/learn\/wp-content\/uploads\/2021\/10\/10-min-scaled.jpg?id=2121) !important;background-position: center !important;background-repeat: no-repeat !important;background-size: cover !important;}” el_id=”bg-sec”][vc_row full_width=”stretch_row_content” el_class=”faaaq”][vc_column width=”1\/12″][\/vc_column][vc_column width=”1\/2″][vc_row_inner content_placement=”middle” el_class=”topheadingg”][vc_column_inner width=”1\/6″][vc_single_image image=”2283″ img_size=”full”][\/vc_column_inner][vc_column_inner width=”5\/6″][vc_column_text]\n<h5>How to Solve Linear Systems of Equations by Substitution<\/h5>\n[\/vc_column_text][\/vc_column_inner][\/vc_row_inner][vc_column_text el_class=”headingnew”]Concept[\/vc_column_text][vc_column_text]In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought the common solutions. A system of equations can be classified in a similar manner as single equations.<\/p>\n<p>Any system of equations can be solved in different methods. To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.<\/p>\n<p><strong>Substitution Method<\/strong><br \/>\n<strong>Elimination Method<\/strong><br \/>\n<strong>Graphical Method<\/strong><\/p>\n<p>For solving the system of equations using the substitution method given two linear equations in x and y, express y in terms x in one of the equations and then substitute it in 2nd equation.[\/vc_column_text][vc_column_text el_class=”headingnew”]Rules[\/vc_column_text][vc_column_text]<strong>Step 1:<\/strong> Solve one of the equations for one of the variables.<\/p>\n<p><strong>Step 2:<\/strong> Substitute the expression of the isolated variable in the other linear equation.<\/p>\n<p><strong>Step 3:<\/strong> Solve the equation, and you have one of the coordinates of the intersection.<\/p>\n<p><strong>Step 4:<\/strong> Then plug in the value of the coordinate found in step 3 to either equation to find the other coordinate.[\/vc_column_text][vc_column_text el_class=”headingnew”]Example[\/vc_column_text][vc_column_text]<strong><span style=\"color: #16a085;\">Solve this system of equations using the substitution method.<\/span><\/strong><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-7x-2y=-13\" alt=\"\\fn_phv -7x-2y=-13\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 1px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x-2y=11\" alt=\"\\fn_phv x-2y=11\" align=\"absmiddle\" \/><br \/>\n[\/vc_column_text][vc_column_text el_class=”headingnew”]Solution[\/vc_column_text][vc_column_text]First, make one equation equal to a variable. It is easiest to make the second equation equal to x. Then, rewrite the first equation replacing the variable.<br \/>\n<img style=\"margin-top: 16px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;-&space;2y&space;=&space;11\" alt=\"\\fn_phv x - 2y = 11\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 0px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;=&space;11&space;+&space;2y\" alt=\"\\fn_phv x = 11 + 2y\" align=\"absmiddle\" \/><\/p>\n<p>Substitute the value of x in the other linear equation and solve it.<br \/>\n<img style=\"margin-top: 16px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-7x&space;-&space;2y&space;=&space;-13\" alt=\"\\fn_phv -7x - 2y = -13\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-7(11&space;+&space;2y)&space;-&space;2y&space;=&space;-13\" alt=\"\\fn_phv -7(11 + 2y) - 2y = -13\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-77&space;-&space;14y&space;-&space;2y&space;=&space;-13\" alt=\"\\fn_phv -77 - 14y - 2y = -13\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-77&space;-&space;16y&space;=&space;-13\" alt=\"\\fn_phv -77 - 16y = -13\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;-16y&space;=&space;64\" alt=\"\\fn_phv -16y = 64\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 0px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;y&space;=&space;-4\" alt=\"\\fn_phv y = -4\" align=\"absmiddle\" \/><\/p>\n<p>Then, plug that value into one of the original equations.<br \/>\n<img style=\"margin-top: 16px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;=&space;11&space;+&space;2y\" alt=\"\\fn_phv x = 11 + 2y\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;=&space;11&space;+&space;2(-4)\" alt=\"\\fn_phv x = 11 + 2(-4)\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;=&space;11&space;-&space;8\" alt=\"\\fn_phv x = 11 - 8\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 0px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;=&space;3\" alt=\"\\fn_phv x = 3\" align=\"absmiddle\" \/><\/p>\n<p>The solution is <span style=\"color: #ff0000;\"><strong>(3, -4)<\/strong><\/span>[\/vc_column_text][vc_column_text el_class=”headingnew”]Practice How to Solve Linear Systems of Equations by Substitution[\/vc_column_text][vc_row_inner el_class=”boxxborder”][vc_column_inner][vc_column_text]\n<h5 style=\"margin-bottom: 10px;\">Practice Problem 1<\/h5>\n<p><strong><span style=\"color: #16a085;\">Solve this system of equations using the substitution method.<\/span><\/strong><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;x&space;-&space;y&space;=&space;3\" alt=\"\\fn_phv x - y = 3\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;y&space;=&space;3x&space;-&space;1\" alt=\"\\fn_phv y = 3x - 1\" align=\"absmiddle\" \/><br \/>\n<a href=\"https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution\"><img loading=\"lazy\" class=\"alignnone wp-image-6005 size-full\" style=\"margin-top: 10px; margin-bottom: 1px;\" src=\"https:\/\/oryxlearning.com\/learn\/wp-content\/uploads\/2021\/12\/table-temp.png\" alt=\"\" width=\"151\" height=\"55\" \/><\/a><br \/>\n[\/vc_column_text][vc_single_image image=”2159″ img_size=”full” alignment=”right” onclick=”custom_link” link=”https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution”][\/vc_column_inner][\/vc_row_inner][vc_empty_space][vc_row_inner el_class=”boxxborder”][vc_column_inner][vc_column_text]\n<h5 style=\"margin-bottom: 10px;\">Practice Problem 2<\/h5>\n<p><strong><span style=\"color: #16a085;\">Solve this system of equations using the substitution method.<\/span><\/strong><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;4x&space;-&space;5y&space;=&space;-23\" alt=\"\\fn_phv 4x - 5y = -23\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;2x&space;+&space;y&space;=&space;-1\" alt=\"\\fn_phv 2x + y = -1\" align=\"absmiddle\" \/><br \/>\n<a href=\"https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution\"><img loading=\"lazy\" class=\"alignnone wp-image-6005 size-full\" style=\"margin-top: 10px; margin-bottom: 1px;\" src=\"https:\/\/oryxlearning.com\/learn\/wp-content\/uploads\/2021\/12\/table-temp.png\" alt=\"\" width=\"151\" height=\"55\" \/><\/a><br \/>\n[\/vc_column_text][vc_single_image image=”2159″ img_size=”full” alignment=”right” onclick=”custom_link” link=”https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution”][\/vc_column_inner][\/vc_row_inner][vc_empty_space][vc_row_inner el_class=”boxxborder”][vc_column_inner][vc_column_text]\n<h5 style=\"margin-bottom: 10px;\">Practice Problem 3<\/h5>\n<p><strong><span style=\"color: #16a085;\">A movie theater charges $5 for adults and $3 for children. Last Friday, 700 tickets were sold to the public totaling $2,900. Given the two equations:<\/span><\/strong><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;a&space;+&space;c&space;=&space;700\" alt=\"\\fn_phv a + c = 700\" align=\"absmiddle\" \/><br \/>\n<img style=\"margin-top: 11px; margin-bottom: 10px;\" src=\"https:\/\/latex.codecogs.com\/svg.latex?\\fn_phv&space;5a&space;+&space;3c&space;=&space;2,900\" alt=\"\\fn_phv 5a + 3c = 2,900\" align=\"absmiddle\" \/><br \/>\nFind how many adults and how many children attended the movie.<br \/>\n<a href=\"https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution\"><img loading=\"lazy\" class=\"alignnone wp-image-6011 size-full\" style=\"margin-top: 10px; margin-bottom: 1px;\" src=\"https:\/\/oryxlearning.com\/learn\/wp-content\/uploads\/2021\/12\/option-with-input.png\" alt=\"\" width=\"276\" height=\"77\" \/><\/a>[\/vc_column_text][vc_single_image image=”2159″ img_size=”full” alignment=”right” onclick=”custom_link” link=”https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-substitution”][\/vc_column_inner][\/vc_row_inner][vc_empty_space][\/vc_column][vc_column width=”1\/3″ css=”.vc_custom_1633369011398{margin-bottom: 50px !important;}”][vc_row_inner el_class=”stickynotee”][vc_column_inner][vc_single_image image=”2265″ img_size=”full” alignment=”center” css_animation=”fadeIn” css=”.vc_custom_1633509989314{border-radius: 10px !important;}” el_class=”stickypin”][vc_column_text]A <strong>system of linear equations<\/strong> – is a set of equations which are satisfied by the same set of variables.<\/p>\n<p><strong>Variable<\/strong> – A symbol, usually a letter, used to represent a number in mathematical expressions or sentences.<\/p>\n<p><strong>Equation<\/strong> – A mathematical sentence stating that two quantities are equal.<\/p>\n<p><strong>Slope intercept form<\/strong> \u2013 a linear equation written in the form <strong>y = mx + b<\/strong>, where <strong>m<\/strong> stands for the slope, and <strong>b<\/strong> stands for the y-intercept.<\/p>\n<p><strong>Standard form<\/strong> \u2013 a linear equation written in the form <strong>Ax + By = C<\/strong><\/p>\n<p><strong>Solution<\/strong> – A solution to a system of equations is the value of the variables that makes ALL the equations True.<\/p>\n<p><strong>Substitution<\/strong> – Putting numbers in place of letters to calculate the value of an expression.[\/vc_column_text][\/vc_column_inner][\/vc_row_inner][vc_column_text el_class=”boxxborder2″]<span style=\"color: #ed5600;\"><strong>Pre-requisite Skills<\/strong><\/span><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/solve-equations-with-rational-coefficients\"><span style=\"color: #ed5600;\">Solve Equations with Rational Coefficients<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/solve-two-step-equations\"><span style=\"color: #ed5600;\">Solve Two-Step Equations<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/write-two-step-equations\"><span style=\"color: #ed5600;\">Write Two-Step Equations<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/Solve%20Two-Sided%20Equations%20with%20Rational%20Coefficients\"><span style=\"color: #ed5600;\">Solve Two-Sided Equations with Rational Coefficients<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/solve-multi-step-equations\"><span style=\"color: #ed5600;\">Solve Multi-Step Equations<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/slope-intercept-form-of-an-equation\"><span style=\"color: #ed5600;\">Slope-intercept Form of an Equation<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/graphing-lines-using-intercepts\"><span style=\"color: #ed5600;\">Graphing Lines Using Intercepts<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/write-linear-equations\"><span style=\"color: #ed5600;\">Write Linear Equations<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/direct-variation-equations-y-mx-\"><span style=\"color: #ed5600;\">Direct Variation Equations (y = mx)<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/solutions-of-systems-of-equations\"><span style=\"color: #ed5600;\">Solutions of Systems of Equations<\/span><\/a>[\/vc_column_text][vc_column_text el_class=”boxxborder2″ css=”.vc_custom_1638350516191{border-top-width: 7px !important;border-right-width: 7px !important;border-bottom-width: 7px !important;border-left-width: 7px !important;background-position: center !important;background-repeat: no-repeat !important;background-size: contain !important;border-left-color: #333399 !important;border-left-style: solid !important;border-right-color: #333399 !important;border-right-style: solid !important;border-top-color: #333399 !important;border-top-style: solid !important;border-bottom-color: #333399 !important;border-bottom-style: solid !important;border-radius: 35px !important;}”]<span style=\"color: #333399;\"><strong>Related Skills<\/strong><\/span><br \/>\n<a href=\"https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-systems-of-equations-by-elimination\"><span style=\"color: #333399;\">Solve Systems of Equations by Elimination<\/span><\/a><br \/>\n<a href=\"https:\/\/oryxlearning.com\/question\/maths\/8th-grade\/solve-linear-systems-of-equations-by-graphing\"><span style=\"color: #333399;\">Solve Linear Systems of Equations by Graphing<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/applications-of-systems\"><span style=\"color: #333399;\">Applications of Systems<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/representing-relationships\"><span style=\"color: #333399;\">Representing Relationships<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/relations\"><span style=\"color: #333399;\">Relations<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/functions\"><span style=\"color: #333399;\">Functions<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/linear-functions\"><span style=\"color: #333399;\">Linear Functions<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/linear-and-nonlinear-functions\"><span style=\"color: #333399;\">Linear and Nonlinear Functions<\/span><\/a><br \/>\n<a href=\"https:\/\/en.oryxlearning.com\/question\/maths\/8th-grade\/quadratic-functions\"><span style=\"color: #333399;\">Quadratic Functions<\/span><\/a><br 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!important;background-position: center !important;background-repeat: no-repeat !important;background-size: cover !important;}” el_id=”bg-sec”][vc_row full_width=”stretch_row_content” el_class=”faaaq”][vc_column width=”1\/12″][\/vc_column][vc_column width=”1\/2″][vc_row_inner content_placement=”middle” el_class=”topheadingg”][vc_column_inner width=”1\/6″][vc_single_image image=”2283″ img_size=”full”][\/vc_column_inner][vc_column_inner width=”5\/6″][vc_column_text] How to Solve Linear Systems of Equations by Substitution [\/vc_column_text][\/vc_column_inner][\/vc_row_inner][vc_column_text el_class=”headingnew”]Concept[\/vc_column_text][vc_column_text]In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO 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