758 CHAPTER 11 Systems of Equations and Inequalities 1 Solve Systems of Equations by Substitution Most of the time we use algebraic methods to obtain exact solutions. A number of methods are available for solving systems of linear equations algebraically. In this section, we introduce two methods: substitution and elimination. We illustrate the method of substitution by solving the system given in Example 3. How to Solve a System of Linear Equations by Substitution Solve: + =− − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 1 4 6 42 (1) (2) EXAMPLE 4 Figure 3 Step-by-Step Solution Step 1 Pick one of the equations, and solve for one variable in terms of the remaining variable(s). Solve equation (1) for y. + =− = − − x y y x 2 1 2 1 Equation (1) Step 2 Substitute the result into the remaining equation(s). Substitute − −x2 1 for y in equation (2).The result is an equation containing just the variable x, which we can solve. ( ) − + = − + − − = x y x x 4 6 42 4 6 2 1 42 Step 4 Find the values of the remaining variables by back-substitution. Because we know that = − x 3, we can find the value of y by back-substitution, that is, by substituting −3 for x in one of the original equations. Equation (1) seems easier to work with, so we will back-substitute into equation (1). ( ) + =− − + =− − + =− = x y y y y 2 1 2 3 1 6 1 5 Step 3 If one equation in one variable results, solve this equation. Otherwise, repeat Steps 1 and 2 until a single equation with one variable remains. − − − = − − = − = = − x x x x x 4 12 6 42 16 6 42 16 48 3 Step 5 Check the solution found. We have = − x 3 and = y 5. Verify that both equations are satisfied (true) for these values. ( ) ( ) + =− − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ − + = − + = − − − + ⋅ = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 1 4 6 42 2 3 5 6 5 1 43 65 12 30 42 The solution of the system is = − x 3 and = y 5. The solution can also be written as the ordered pair ( ) −3, 5 . The solution, since it is the point of intersection of the two lines, can be verified using Desmos, as shown in Figure 3. Equation (1) Substitute 3− for x. Simplify. Solve for y. Distribute. Combine like terms. Add 6 to both sides. Solve for x. Now Use Substitution to Work PROBLEM 21 2 Solve Systems of Equations by Elimination A second method for solving a system of linear equations is the method of elimination. This method is usually preferred over substitution if substitution leads to fractions or if the system contains more than two variables. Elimination also provides the motivation for solving systems using matrices (the subject of Section 11.2). Equation (2) Substitute x2 1 − − for y in (2).
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