6.7 Solving Systems of Linear Equations 355 Substitution Method SOLVING A SYSTEM OF EQUATIONS BY THE SUBSTITUTION METHOD 1. Solve one of the equations for one of the variables. If possible, solve for a variable with a numerical coefficient of 1. By doing so, you may avoid working with fractions. 2. Substitute the expression determined in Step 1 into the other equation. This step yields an equation in terms of a single variable. 3. Solve the equation determined in Step 2 for the variable. 4. Substitute the value determined in Step 3 into the equation you rewrote in Step 1 and solve for the remaining variable. PROCEDURE Examples 3, 4, and 5 illustrate the substitution method . Example 3 A Unique Solution by the Substitution Method Solve the following system of equations by substitution. x y x y 4 2 1 + = − = − Solution The numerical coefficients of the x- and y-terms in the equation x y 4 + = are both 1. Thus, we can solve this equation for either x or y. Let’s solve for x in the first equation. STEP 1. + = + − = − = − x y x y y y x y 4 4 4 Subtract y from both sides of the equations. STEP 2. Substitute y 4 − for x in the second equation. − = − − − = − x y y y 2 1 2(4 ) 1 STEP 3. Now solve the equation for y. − − = − − = − − − = − − − = − − − = − − = y y y y y y y 8 2 1 8 3 1 8 8 3 1 8 3 9 3 3 9 3 3 Distributive property Subtract 8 from both sides of the equation. Divide both sides of the equation by 3− . STEP 4. Substitute y 3 = in the equation solved for x and determine the value of x. x y x x 4 4 3 1 = − = − = Thus, the solution is the ordered pair (1, 3). 7 Now try Exercise 25 Timely Tip When solving a system of equations, once you successfully solve for one of the variables, make sure you solve for the other variable. Remember that a solution to a system of equations must contain a numerical value for each variable in the system.
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