Section 4.2 Solving a System of Equations in Two Variables by the Substitution Method

The Substitution Method Procedure for Solving a System of Equations by the Substitution Method Solve one of the two equations for one variable. If possible, solve for a variable with a coefficient of 1 or 1. Substitute the expression obtained in Step 1 for this variable in the other equation. You now have one equation with one variable. Solve this equation to find the value for that one variable. Substitute this value for the variable into one of the original equations to obtain a value for the other variable. Check the solution in each equation to verify the results. 2

Example Find the solution. 2x – y = 13 – 4x – 9y = 7 y = 2x – 13 This variable is the easiest to isolate. y = 2x – 13 Solve the first equation for y. – 4x – 9y = 7 This is the original second equation. – 4x – 9(2x – 13) = 7 Substitute the expression into the other equation and solve. – 4x – 18x + 117 = 7 – 22x = – 110 x = 5 Substitute this value into one of the original equations. Continued 3

Example (cont) 2x – y = 13 – 4x – 9y = 7 2x – y = 13 2(5) – y = 13 First equation Second equation 2x – y = 13 2(5) – y = 13 Substitute x = 5 into the first equation. 10 – y = 13 Solve for y. –y = 3 y = –3 The solution is (5, –3). Continued 4

Example (cont) Check the solution (5, –3) in both original equations. 2x – y = 13 – 4x – 9y = 7 2(5) – (3) = 13 – 4(5) – 9(3) = 7 10 + 3 = 13 – 20 + 27 = 7 13 = 13  7 = 7  5

Example Find the solution. Clear the fractions from the equation. Multiply each term by the LCD. Solve for y. Substitute into the other equation. Continued Simplify. 6

Example (cont) Combine like terms and simplify. Substitute this value into one of the original equations. The solution is (4, 1). Continued 7

Example (cont) Check the solution (–4, –1) in both original equations.   8