The Substitution Method Find the solution. 2x – y = 13 – 4x – 9y = 7 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 1

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 2

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  3

The Addition Method Solve by addition. 5x – 3y = 14 2x – y = 6 Multiply each term of equation (2) by 3.  6x + 3y = 18 This equation is equivalent to equation (2). 5x – 3y = 14  6x + 3y = 18  x =  4 Add the two equations. Substitute this value into either equation to find y. x = 4 Continued 4

Example (cont) 5x – 3y = 14 2x – y = 6 2x – y = 6 2(4) – y = 6 Substitute. 8  y = 6 y = 2 The solution is (4, 2). y = 2 5

Example Solve. To eliminate the variable y, we want the coefficients of the y terms to be opposites. 6

Example The solution is (–1, 2) and it checks. Multiply first equation by 3. Multiply the second equation by 5. Solve for x. Replace x by –1 in equations (3), and solve for y. The solution is (–1, 2) and it checks. 7