{"id":2235,"date":"2026-05-18T02:52:12","date_gmt":"2026-05-18T02:52:12","guid":{"rendered":"https:\/\/www.think10x.ai\/blog\/?p=2235"},"modified":"2026-05-18T02:52:13","modified_gmt":"2026-05-18T02:52:13","slug":"linear-algebraic-equations","status":"publish","type":"post","link":"https:\/\/www.think10x.ai\/blog\/linear-algebraic-equations\/","title":{"rendered":"How to Understand Linear Algebraic Equations Step-by-Step"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"2235\" class=\"elementor elementor-2235\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5a83628 e-flex e-con-boxed e-con e-parent\" data-id=\"5a83628\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-43c556a elementor-alert-info elementor-widget elementor-widget-alert\" data-id=\"43c556a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"alert.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-alert\" role=\"alert\">\n\n\t\t\t\t\t\t<span class=\"elementor-alert-title\">Quick Answer<\/span>\n\t\t\t\n\t\t\t\t\t\t<span class=\"elementor-alert-description\">A linear algebraic equation is written in the form Ax + B = C, where every variable has a power of 1. To solve it, simplify both sides, move variable terms to one side and constants to the other, divide to isolate the variable, then substitute your answer back to verify it.<\/span>\n\t\t\t\n\t\t\t\t\t\t<button type=\"button\" class=\"elementor-alert-dismiss\" aria-label=\"Dismiss this alert.\">\n\t\t\t\t\t\t\t\t\t<span aria-hidden=\"true\">&times;<\/span>\n\t\t\t\t\t\t\t<\/button>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-86d4918 e-flex e-con-boxed e-con e-parent\" data-id=\"86d4918\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-33cfb05 elementor-widget elementor-widget-text-editor\" data-id=\"33cfb05\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>What Are Linear Algebraic Equations?<\/h2><p><span style=\"font-weight: 400;\">The word &#8220;linear&#8221; traces back to the Latin <\/span><i><span style=\"font-weight: 400;\">linearis<\/span><\/i><span style=\"font-weight: 400;\">, meaning &#8220;belonging to a line,&#8221; itself derived from <\/span><i><span style=\"font-weight: 400;\">linea<\/span><\/i><span style=\"font-weight: 400;\">, meaning &#8220;thread or string.&#8221; When a linear algebraic equation is graphed, it always produces a straight line, because no variable is raised above a power of 1. No squares, no cubes, no roots.<\/span><\/p><p><span style=\"font-weight: 400;\">The standard form is <\/span><span style=\"font-weight: 400;\"><strong>Ax<\/strong> + <strong>B<\/strong> = <strong>C<\/strong><\/span><\/p><p><strong>A<\/strong><span style=\"font-weight: 400;\"> is the coefficient (the number multiplying the variable), <\/span><strong>x<\/strong><span style=\"font-weight: 400;\"> is the variable (the unknown you are solving for), <\/span><strong>B<\/strong><span style=\"font-weight: 400;\"> is a constant added or subtracted, and <\/span><strong>C<\/strong><span style=\"font-weight: 400;\"> is the value on the right-hand side.<\/span><\/p><h3>Example<\/h3><p><span style=\"font-weight: 400;\">3x + 5 = 14<\/span><\/p><p><span style=\"font-weight: 400;\">The variable <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> has no exponent other than 1, and solving it gives exactly one answer.<\/span><\/p><p><span style=\"font-weight: 400;\">Compare that to <\/span><span style=\"font-weight: 400;\">x\u00b2 + 5 = 14<\/span><span style=\"font-weight: 400;\">. The moment an exponent of 2 appears, it becomes a quadratic equation, not a linear one.<\/span><\/p><p><span style=\"font-weight: 400;\">A <\/span><b>linear algebraic expression<\/b><span style=\"font-weight: 400;\"> like <\/span><span style=\"font-weight: 400;\">3x + 5<\/span><span style=\"font-weight: 400;\"> has no equals sign. Add an equals sign and a value on the right and it becomes a solvable linear equation.<\/span><\/p><h2>Why Linear Equations Matter<\/h2><p><span style=\"font-weight: 400;\">Linear algebra is the mathematical foundation behind machine learning, engineering, economics, and data science. Before you can work with matrices or systems of equations, you need a solid grip on algebra linear equations. Every advanced concept in linear algebra builds from solving equations like <\/span><span style=\"font-weight: 400;\">Ax + B = C<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><h3>Real-world uses of linear equations<\/h3>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-a502562 e-flex e-con-boxed e-con e-parent\" data-id=\"a502562\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-83a4d90 eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"83a4d90\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"83a4d90\" id=\"eael-data-table-wrapper-83a4d90\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-83a4d90\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Application<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Example<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tBudgeting\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIf you already have $200 and save $50 each week, how many weeks will it take to reach $600?\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tSpeed and distance\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tWhen do two vehicles travelling toward each other meet?\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tEngineering\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tCalculating load forces on a structure\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tBusiness\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tFinding the break-even point where revenue equals total costs\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tData science\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tLinear regression, the core of predictive modelling\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c0fb02a e-flex e-con-boxed e-con e-parent\" data-id=\"c0fb02a\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-82ea7fb elementor-widget elementor-widget-text-editor\" data-id=\"82ea7fb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>Key Terms to Know Before You Start<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-9a2ea63 e-flex e-con-boxed e-con e-parent\" data-id=\"9a2ea63\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f327107 eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"f327107\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"f327107\" id=\"eael-data-table-wrapper-f327107\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-f327107\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Term<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Definition<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Example<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tVariable\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tAn unknown value represented by a letter\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx, y, z\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tCoefficient\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tThe number multiplying a variable\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIn 5x, the coefficient is 5\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tConstant\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tA fixed number with no variable attached\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIn 3x + 7, the constant is 7\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tLinear algebraic expression\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tVariables and constants combined, no equals sign\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t4x + 2\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tLinear algebraic equation\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tAn expression set equal to a value\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t4x + 2 = 10\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tSolution\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tThe value of the variable that makes the equation true\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx = 2\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tInverse operation\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tThe operation that undoes another\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tSubtracting undoes addition\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-3bed46b e-flex e-con-boxed e-con e-parent\" data-id=\"3bed46b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4515efc elementor-widget elementor-widget-text-editor\" data-id=\"4515efc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A linear expression in algebra is any combination of variables and constants where no variable has a power above 1, for example <\/span><span style=\"font-weight: 400;\">6x &#8211; 4<\/span><span style=\"font-weight: 400;\"> or <\/span><span style=\"font-weight: 400;\">2x + 3y<\/span><span style=\"font-weight: 400;\">. Add an equals sign and it becomes a solvable equation.<\/span><\/p><h2>Types of Linear Algebraic Equations<\/h2><h3>1. One-variable linear equations<\/h3><p><span style=\"font-weight: 400;\">Form <\/span><span style=\"font-weight: 400;\">Ax + B = C,<\/span><span style=\"font-weight: 400;\"> example <\/span><span style=\"font-weight: 400;\">3x + 5 = 14<\/span><span style=\"font-weight: 400;\"> has exactly one solution.<\/span><\/p><h3>2. Two-variable linear equations<\/h3><p><span style=\"font-weight: 400;\">Form <\/span><span style=\"font-weight: 400;\">Ax + By = C,<\/span><span style=\"font-weight: 400;\"> example <\/span><span style=\"font-weight: 400;\">2x + 3y = 12<\/span><span style=\"font-weight: 400;\"> produces infinite solutions on its own. A second equation is needed to find a unique answer, which is the basis of systems of linear equations.<\/span><\/p><h3>3. Forms of a linear equation<\/h3><p><span style=\"font-weight: 400;\">The standard form in two variables is <\/span><span style=\"font-weight: 400;\">Ax + By = C<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">The slope-intercept form is <\/span><span style=\"font-weight: 400;\">y = mx + b<\/span><span style=\"font-weight: 400;\">, where <\/span><span style=\"font-weight: 400;\">m<\/span><span style=\"font-weight: 400;\"> is the slope (rate of change) and <\/span><span style=\"font-weight: 400;\">b<\/span><span style=\"font-weight: 400;\"> is the y-intercept. This form is most commonly used when graphing linear function algebra problems.<\/span><\/p><h3>4. Types by solution behaviour<\/h3>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-19bcd54 e-flex e-con-boxed e-con e-parent\" data-id=\"19bcd54\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ffe442d eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"ffe442d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"ffe442d\" id=\"eael-data-table-wrapper-ffe442d\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-ffe442d\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Type<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">What It Means<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Example<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tConditional\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tExactly one solution\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t2x + 3 = 11 gives x = 4\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIdentity\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tInfinite solutions, always true\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t2x + 4 = 2(x + 2)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tInconsistent\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tNo solution, a contradiction\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t2x + 3 = 2x + 7\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d37a6ff e-flex e-con-boxed e-con e-parent\" data-id=\"d37a6ff\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-137765a elementor-widget elementor-widget-text-editor\" data-id=\"137765a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">When solving produces a result like <\/span><span style=\"font-weight: 400;\">3 = 7<\/span><span style=\"font-weight: 400;\">, the equation is inconsistent. No value of <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> will ever make it true.<\/span><\/p><h2>How to Solve a One-Variable Linear Equation Step by Step<\/h2><h3>Question<\/h3><p><span style=\"font-weight: 400;\">5x &#8211; 3 = 2x + 9<\/span><\/p><h3>Step 1. Simplify both sides<\/h3><p><span style=\"font-weight: 400;\">Check both sides for parentheses to expand and like terms to combine.<\/span><\/p><p><span style=\"font-weight: 400;\">5x &#8211; 3 = 2x + 9<\/span><span style=\"font-weight: 400;\"> is already simplified. If the equation were <\/span><span style=\"font-weight: 400;\">3(x + 2) + 2x = 9<\/span><span style=\"font-weight: 400;\">, you would expand first to get <\/span><span style=\"font-weight: 400;\">3x + 6 + 2x = 9<\/span><span style=\"font-weight: 400;\">, then <\/span><span style=\"font-weight: 400;\">5x + 6 = 9<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><p>Tip &#8211;<span style=\"font-weight: 400;\">\u00a0Move the variable with the smaller coefficient across to avoid working with negative coefficients.<\/span><\/p><h3>Step 2. Move all variable terms to one side<\/h3><p><span style=\"font-weight: 400;\">Subtract <\/span><span style=\"font-weight: 400;\">2x<\/span><span style=\"font-weight: 400;\"> from both sides.<\/span><\/p><ul><li><span style=\"font-weight: 400;\">5x &#8211; 3 &#8211; 2x = 2x + 9 &#8211; 2x<\/span><\/li><li><span style=\"font-weight: 400;\">3x &#8211; 3 = 9<\/span><\/li><\/ul><h3>Step 3. Move all constants to the other side<\/h3><p><span style=\"font-weight: 400;\">Add <\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> to both sides.<\/span><\/p><ul><li><span style=\"font-weight: 400;\">3x &#8211; 3 + 3 = 9 + 3<\/span><\/li><li><span style=\"font-weight: 400;\">3x = 12<\/span><\/li><\/ul><h3>Step 4. Isolate the variable<\/h3><p><span style=\"font-weight: 400;\">Divide both sides by <\/span><span style=\"font-weight: 400;\">3.<\/span><\/p><ul><li><span style=\"font-weight: 400;\">3x \u00f7 3 = 12 \u00f7 3<\/span><\/li><li><span style=\"font-weight: 400;\">x = 4<\/span><\/li><\/ul><h3>Step 5. Verify your answer<\/h3><p><span style=\"font-weight: 400;\">Substitute <\/span><span style=\"font-weight: 400;\">x = 4<\/span><span style=\"font-weight: 400;\"> back into the original equation.<\/span><\/p><ul><li><span style=\"font-weight: 400;\">5(4) &#8211; 3 = 2(4) + 9<\/span><\/li><li><span style=\"font-weight: 400;\">20 &#8211; 3 = 8 + 9<\/span><\/li><li><span style=\"font-weight: 400;\">17 = 17<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Both sides are equal. The answer is correct.<\/span><\/p><p><b>Never skip verification.<\/b><span style=\"font-weight: 400;\"> It is the only way to catch an error made in any earlier step.<\/span><\/p><h2>How to Solve Linear Equations with Fractions<\/h2><h3>Question<\/h3><p><span style=\"font-weight: 400;\">x\/2 + 3 = 7<\/span><\/p><h3>Solution<\/h3><h4>Step 1.<span style=\"font-weight: 400;\"> Eliminate the fraction by multiplying every term on both sides by the Least Common Denominator (LCD) <\/span><\/h4><p><span style=\"font-weight: 400;\">The LCD here is <\/span><span style=\"font-weight: 400;\">2 &#8211;<\/span><\/p><ul><li><span style=\"font-weight: 400;\">2(x\/2) + 2(3) = 2(7)<\/span><\/li><li><span style=\"font-weight: 400;\">x + 6 = 14<\/span><\/li><\/ul><h4>Step 2.<span style=\"font-weight: 400;\"> Solve the equation<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">x = 14 &#8211; 6<\/span><\/li><li><span style=\"font-weight: 400;\">x = 8<\/span><\/li><\/ul><h4>Step 3.<span style=\"font-weight: 400;\"> Verify<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">8\/2 + 3 = 7<\/span><\/li><li><span style=\"font-weight: 400;\">4 + 3 = 7<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">The most common error here is multiplying only the fraction by the LCD and forgetting to apply it to every other term on both sides.<\/span><\/p><h2>How to Solve Systems of Linear Equations<\/h2><p><span style=\"font-weight: 400;\">When solving systems of linear equations algebraically, two methods are available.<\/span><\/p><h3>The substitution method<\/h3><p><span style=\"font-weight: 400;\">Best used when one variable is easy to isolate.<\/span><\/p><p><span style=\"font-weight: 400;\">x + y = 10 &#8211; equation (1)<\/span><\/p><p><span style=\"font-weight: 400;\">2x &#8211; y = 5 &#8211;\u00a0 equation (2)<\/span><\/p><h4>Step 1.<span style=\"font-weight: 400;\"> Isolate one variable from equation (1)<\/span><\/h4><p><span style=\"font-weight: 400;\">x = 10 &#8211; y<\/span><\/p><h4>Step 2.<span style=\"font-weight: 400;\"> Substitute into equation (2)<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">2(10 &#8211; y) &#8211; y = 5<\/span><\/li><li><span style=\"font-weight: 400;\">20 &#8211; 2y &#8211; y = 5<\/span><\/li><li><span style=\"font-weight: 400;\">20 &#8211; 3y = 5<\/span><\/li><li><span style=\"font-weight: 400;\">-3y = -15<\/span><\/li><li><span style=\"font-weight: 400;\">y = 5<\/span><\/li><\/ul><h4>Step 3.<span style=\"font-weight: 400;\">\u00a0Substitute <\/span><span style=\"font-weight: 400;\">y = 5<\/span><span style=\"font-weight: 400;\"> back into equation (1)<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">x + 5 = 10<\/span><\/li><li><span style=\"font-weight: 400;\">x = 5<\/span><\/li><\/ul><h4>Step 4.<span style=\"font-weight: 400;\"> Verify in both equations<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">5 + 5 = 10<\/span><\/li><li><span style=\"font-weight: 400;\">2(5) &#8211; 5 = 5<\/span><\/li><\/ul><p>Solution is\u00a0<span style=\"font-weight: 400;\">x = 5, y = 5<\/span><\/p><h3>The elimination method<\/h3><p><span style=\"font-weight: 400;\">Best used when coefficients can be matched. This is the faster approach when both equations are already in standard form.<\/span><\/p><p><span style=\"font-weight: 400;\">3x + 2y = 16 &#8211; equation (1)<\/span><\/p><p><span style=\"font-weight: 400;\">x &#8211; 2y = 0 &#8211; equation (2)<\/span><\/p><h4>Step 1.<span style=\"font-weight: 400;\">\u00a0Add the two equations to eliminate <\/span><span style=\"font-weight: 400;\">y<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">(3x + 2y) + (x &#8211; 2y) = 16 + 0<\/span><\/li><li><span style=\"font-weight: 400;\">4x = 16<\/span><\/li><li><span style=\"font-weight: 400;\">x = 4<\/span><\/li><\/ul><h4>Step 2.<span style=\"font-weight: 400;\">\u00a0Substitute <\/span><span style=\"font-weight: 400;\">x = 4<\/span><span style=\"font-weight: 400;\"> into equation (2)<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">4 &#8211; 2y = 0<\/span><\/li><li><span style=\"font-weight: 400;\">2y = 4<\/span><\/li><li><span style=\"font-weight: 400;\">y = 2<\/span><\/li><\/ul><h4>Step 3.<span style=\"font-weight: 400;\"> Verify<\/span><\/h4><ul><li><span style=\"font-weight: 400;\">3(4) + 2(2) = 16<\/span><\/li><li><span style=\"font-weight: 400;\">4 &#8211; 2(2) = 0<\/span><\/li><\/ul><p>Solution is\u00a0<span style=\"font-weight: 400;\">x = 4, y = 2<\/span><\/p><h2>Substitution vs Elimination<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-1897f0f e-flex e-con-boxed e-con e-parent\" data-id=\"1897f0f\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d75320b eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"d75320b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"d75320b\" id=\"eael-data-table-wrapper-d75320b\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-d75320b\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Method<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Use When<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Strength<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Limitation<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tSubstitution\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tOne variable is easy to isolate\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIntuitive and clear\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tGets messy with large coefficients\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tElimination\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tCoefficients already match or can be matched\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tFast in standard form\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tMay require multiplying equations first\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-a5a5f39 e-flex e-con-boxed e-con e-parent\" data-id=\"a5a5f39\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-befbc7b elementor-widget elementor-widget-text-editor\" data-id=\"befbc7b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Both produce the same answer. Choose whichever is faster for the equation in front of you.<\/span><\/p><h2>5 Common Mistakes and How to Avoid Them<\/h2><ol><li><b>Not applying the operation to both sides.<\/b><span style=\"font-weight: 400;\"> If you subtract 3 from the left, subtract 3 from the right. Every time.<\/span><\/li><li><b>Sign errors when moving terms.<\/b><span style=\"font-weight: 400;\"> When a term crosses the equals sign, its sign flips. <\/span><span style=\"font-weight: 400;\">+3<\/span><span style=\"font-weight: 400;\"> becomes <\/span><span style=\"font-weight: 400;\">-3<\/span><span style=\"font-weight: 400;\">. This causes the majority of wrong answers.<\/span><\/li><li><b>Incorrect distribution.<\/b><span style=\"font-weight: 400;\"> In <\/span><span style=\"font-weight: 400;\">3(x + 4)<\/span><span style=\"font-weight: 400;\">, both <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> and <\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> must be multiplied by 3. The result is <\/span><span style=\"font-weight: 400;\">3x + 12<\/span><span style=\"font-weight: 400;\">, not <\/span><span style=\"font-weight: 400;\">3x + 4<\/span><span style=\"font-weight: 400;\">.<\/span><\/li><li><b>Applying the LCD to the fraction only.<\/b><span style=\"font-weight: 400;\"> When clearing fractions, the LCD must be multiplied across every term on both sides.<\/span><\/li><li><b>Skipping verification.<\/b><span style=\"font-weight: 400;\"> Substituting the answer back into the original equation takes under a minute and is the only reliable way to confirm it is correct.<\/span><\/li><\/ol><h2>Practice Problems<\/h2><p><span style=\"font-weight: 400;\">Work through each one before checking the answer.<\/span><\/p><p><b>Level 1.<\/b>\u00a0<span style=\"font-weight: 400;\">4x + 6 = 22<\/span><\/p><p><span style=\"font-weight: 400;\">Answer &#8211; <\/span><span style=\"font-weight: 400;\">4x = 16<\/span><span style=\"font-weight: 400;\">, so <\/span><b>x = 4<\/b><\/p><p><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">4(4) + 6 = 22<\/span><\/p><p><b>Level 2.<\/b>\u00a0<span style=\"font-weight: 400;\">3(x &#8211; 2) + 4 = 13<\/span><\/p><p><span style=\"font-weight: 400;\">Answer &#8211; <\/span><span style=\"font-weight: 400;\">3x &#8211; 6 + 4 = 13<\/span><span style=\"font-weight: 400;\">, then <\/span><span style=\"font-weight: 400;\">3x &#8211; 2 = 13<\/span><span style=\"font-weight: 400;\">, then <\/span><span style=\"font-weight: 400;\">3x = 15<\/span><span style=\"font-weight: 400;\">, so <\/span><b>x = 5<\/b><\/p><p><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">3(5 &#8211; 2) + 4 = 13<\/span><\/p><p><b>Level 3.<\/b>\u00a0<span style=\"font-weight: 400;\">x\/3 + 4 = 7<\/span><\/p><p><span style=\"font-weight: 400;\">Answer &#8211; Multiply by 3 to get <\/span><span style=\"font-weight: 400;\">x + 12 = 21<\/span><span style=\"font-weight: 400;\">, so <\/span><b>x = 9<\/b><\/p><p><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">9\/3 + 4 = 7<\/span><\/p><p><b>Level 4.<\/b>\u00a0<span style=\"font-weight: 400;\">x + y = 8<\/span><span style=\"font-weight: 400;\"> and <\/span><span style=\"font-weight: 400;\">x &#8211; y = 2<\/span><\/p><p><span style=\"font-weight: 400;\">Answer &#8211; Add to get <\/span><span style=\"font-weight: 400;\">2x = 10<\/span><span style=\"font-weight: 400;\">, so <\/span><span style=\"font-weight: 400;\">x = 5<\/span><span style=\"font-weight: 400;\"> and <\/span><b>y = 3<\/b><\/p><p><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">5 + 3 = 8<\/span><span style=\"font-weight: 400;\">\u00a0and <\/span><span style=\"font-weight: 400;\">5 &#8211; 3 = 2<\/span><\/p><h2>Real-World Examples<\/h2><p><b>Budgeting <\/b>&#8211;<span style=\"font-weight: 400;\"> You earn $2,500 per month, spend $1,800 on fixed costs, and want to save $400. How much can you spend per week on extras?<\/span><\/p><ul><li><span style=\"font-weight: 400;\">4w + 400 = 700<\/span><span style=\"font-weight: 400;\"> gives <\/span><span style=\"font-weight: 400;\">w = $75 per week<\/span><\/li><li><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">4(75) + 400 = 700<\/span><\/li><\/ul><p><b>Speed <\/b>&#8211;<span style=\"font-weight: 400;\">\u00a0Two trains leave cities 300 miles apart, travelling toward each other at 60 mph and 90 mph. When do they meet? <\/span><\/p><ul><li><span style=\"font-weight: 400;\">60t + 90t = 300<\/span><span style=\"font-weight: 400;\"> gives <\/span><span style=\"font-weight: 400;\">t = 2 hours<\/span><\/li><li><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">60(2) + 90(2) = 120 + 180 = 300<\/span><\/li><\/ul><p><b>Break-even <\/b>&#8211;<span style=\"font-weight: 400;\">\u00a0A product costs $12 to make and sells for $20. Fixed costs are $400. How many units are needed to break even? <\/span><\/p><ul><li><span style=\"font-weight: 400;\">20x = 12x + 400<\/span><span style=\"font-weight: 400;\"> gives <\/span><span style=\"font-weight: 400;\">x = 50 units<\/span><\/li><li><span style=\"font-weight: 400;\">Check &#8211; <\/span><span style=\"font-weight: 400;\">20(50) = 12(50) + 400 = 1000<\/span><\/li><\/ul><h2>From Linear Equations to Linear Algebra<\/h2><p><span style=\"font-weight: 400;\">Once you can solve linear equations and systems of linear equations algebraically, the next step is linear algebra.<\/span><\/p><p><span style=\"font-weight: 400;\">Linear algebra extends these foundations into matrices, vectors, and transformations. The systems you solve by hand become systems solved by matrix operations at scale. The logic stays the same. The tools get larger.<\/span><\/p><h2>Frequently Asked Questions<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-7de1537 e-flex e-con-boxed e-con e-parent\" data-id=\"7de1537\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-dba89c8 elementor-widget elementor-widget-eael-adv-accordion\" data-id=\"dba89c8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-adv-accordion.default\">\n\t\t\t\t\t            <div class=\"eael-adv-accordion\" id=\"eael-adv-accordion-dba89c8\" data-scroll-on-click=\"no\" data-scroll-speed=\"300\" data-accordion-id=\"dba89c8\" data-accordion-type=\"accordion\" data-toogle-speed=\"300\">\n            <div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-are-linear-equations-in-math\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"1\" aria-controls=\"elementor-tab-content-2301\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What are linear equations in math?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2301\" class=\"eael-accordion-content clearfix\" data-tab=\"1\" aria-labelledby=\"what-are-linear-equations-in-math\"><p><span style=\"font-weight: 400\">Linear equations are algebraic equations where every variable has an exponent of exactly 1. They always produce a straight line when graphed. The standard form is <\/span><span style=\"font-weight: 400\">Ax + B = C<\/span><span style=\"font-weight: 400\"> for one variable and <\/span><span style=\"font-weight: 400\">Ax + By = C<\/span><span style=\"font-weight: 400\"> for two variables.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"-what-is-a-linear-expression-in-algebra\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"2\" aria-controls=\"elementor-tab-content-2302\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\"> What is a linear expression in algebra?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2302\" class=\"eael-accordion-content clearfix\" data-tab=\"2\" aria-labelledby=\"-what-is-a-linear-expression-in-algebra\"><p><span style=\"font-weight: 400\">A linear algebraic expression is a combination of variables and constants with no equals sign, for example <\/span><span style=\"font-weight: 400\">3x + 7<\/span><span style=\"font-weight: 400\">. Add an equals sign and a value on the right and it becomes a linear equation.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-do-you-determine-a-linear-equation\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"3\" aria-controls=\"elementor-tab-content-2303\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How do you determine a linear equation?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2303\" class=\"eael-accordion-content clearfix\" data-tab=\"3\" aria-labelledby=\"how-do-you-determine-a-linear-equation\"><p><span style=\"font-weight: 400\">Look for two things. Every variable must have a power of 1 with no squares or cubes, and there must be an equals sign. If both are true, it is a linear equation.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-do-you-solve-a-system-of-linear-equations-algebraically\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"4\" aria-controls=\"elementor-tab-content-2304\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How do you solve a system of linear equations algebraically?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2304\" class=\"eael-accordion-content clearfix\" data-tab=\"4\" aria-labelledby=\"how-do-you-solve-a-system-of-linear-equations-algebraically\"><p><span style=\"font-weight: 400\">Use substitution (isolate one variable and plug it into the other equation) or elimination (add or subtract equations to cancel a variable). Both methods give the same result.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-do-you-solve-linear-systems-algebraically-when-one-equation-has-fractions\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"5\" aria-controls=\"elementor-tab-content-2305\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How do you solve linear systems algebraically when one equation has fractions?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2305\" class=\"eael-accordion-content clearfix\" data-tab=\"5\" aria-labelledby=\"how-do-you-solve-linear-systems-algebraically-when-one-equation-has-fractions\"><p><span style=\"font-weight: 400\">Multiply every term in the equation by the Least Common Denominator to clear the fractions first, then proceed with substitution or elimination as normal.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-does-no-solution-mean-in-a-linear-equation\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"6\" aria-controls=\"elementor-tab-content-2306\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What does no solution mean in a linear equation?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2306\" class=\"eael-accordion-content clearfix\" data-tab=\"6\" aria-labelledby=\"what-does-no-solution-mean-in-a-linear-equation\"><p><span style=\"font-weight: 400\">It means the equation is inconsistent. The lines are parallel and never meet. Solving produces a false statement like <\/span><span style=\"font-weight: 400\">5 = 9<\/span><span style=\"font-weight: 400\">, which can never be true.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-is-the-difference-between-linear-and-quadratic-equations\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"7\" aria-controls=\"elementor-tab-content-2307\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What is the difference between linear and quadratic equations?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2307\" class=\"eael-accordion-content clearfix\" data-tab=\"7\" aria-labelledby=\"what-is-the-difference-between-linear-and-quadratic-equations\"><p><span style=\"font-weight: 400\">In a linear equation, the highest power of any variable is 1. In a quadratic, it is 2. Linear equations have at most one solution. Quadratic equations can have two.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-are-the-uses-of-linear-equations-in-real-life\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"8\" aria-controls=\"elementor-tab-content-2308\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What are the uses of linear equations in real life?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-2308\" class=\"eael-accordion-content clearfix\" data-tab=\"8\" aria-labelledby=\"what-are-the-uses-of-linear-equations-in-real-life\"><p><span style=\"font-weight: 400\">Budgeting, break-even analysis, speed and distance problems, salary calculations, engineering load calculations, and data science regression models all rely on linear equations.<\/span><b>\u00a0<\/b><\/p><\/div>\n\t\t\t\t\t<\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-76a1046 e-flex e-con-boxed e-con e-parent\" data-id=\"76a1046\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-310e8e5 elementor-widget elementor-widget-text-editor\" data-id=\"310e8e5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>Need Help With Your Own Question?<\/h2><p>Upload a<a href=\"https:\/\/www.think10x.ai\/blog\/take-a-clear-photo-of-a-question\/\">\u00a0clear photo<\/a>\u00a0to Think10x.ai and get an instant video explanation. Built for\u00a0<a href=\"https:\/\/mentomind.ai\/\" rel=\"noopener\">tutors<\/a>\u00a0and\u00a0<a href=\"https:\/\/mentomind.ai\/for-students\/\" rel=\"noopener\">students<\/a>. Private by default.<\/p><p><strong><a href=\"https:\/\/www.think10x.ai\/\">Try\u00a0<\/a><\/strong><a href=\"https:\/\/www.think10x.ai\/studio\"><b>Think10x.ai<\/b><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Quick Answer A linear algebraic equation is written in the form Ax + B = C, where every variable has a power of 1. To solve it, simplify both sides, move variable terms to one side and constants to the other, divide to isolate the variable, then substitute your answer back to verify it. &times; [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":2244,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/media.mentomind.ai\/img\/t10x\/bp\/how_to_understand_linear_algebraic_equations.webp","fifu_image_alt":"Illustration explaining linear algebraic equations step by step with a vector space problem and matrix equations.","footnotes":""},"categories":[8,1],"tags":[],"class_list":["post-2235","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blogs","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2235","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/comments?post=2235"}],"version-history":[{"count":5,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2235\/revisions"}],"predecessor-version":[{"id":2243,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2235\/revisions\/2243"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media\/2244"}],"wp:attachment":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media?parent=2235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/categories?post=2235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/tags?post=2235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}