Solving Systems Using Elimination

Elimination When neither equation is in the slope-intercept form (y =), you can solve the system using elimination. You can add equations to eliminate a variable. Look for like terms that are opposites of each other (will add to zero).

Example 1 5x – 6y = -32 3x + 6y = 48 Make sure like terms are lined up with each other. Look for like terms that are opposites (in this case the –6y and +6y are opposites). Add all the like terms (5x + 3x = 8x) (-6y + 6y = 0) (-32 + 48 = 16)

Example 1 Now you have “eliminated” the y term and have a one-variable equation to solve. 8x = 16 So, x = 2 You have the first half of your ordered pair. Plug in 2 for x in one of the equations to find y. 5(2) – 6y = -32

Example 1 10 – 6y = -32 Subtract 10 from both sides. -6y = -42 Divide both sides by –6 y = 7 Check by replacing x with 2 and y with 7 in the second equation. 3(2) + 6(7) = 48 6 + 42 = 48 That’s true, so the solution is (2,7)

Example 2 x – y = 12 x + y = 22 The y’s are opposites, so add like terms (x + x = 2x, -y + y = 0, and 12 + 22 = 34) Now your equation is 2x = 34. Divide both sides by 2. x = 17

Example 2 Replace x with 17 in the first equation to find y. 17 – y = 12 Subtract 17 from both sides. -y = -5 Divide both sides by –1. y = 5 Replace x with 17 and y with 5 in the second equation to check. 17 + 5 = 22 That is true, so the solution is (17, 5)

Not Always So Easy Sometimes there are not like terms that will add to zero (eliminate). You can multiply or divide all the terms by any number (except zero) to make opposites.

Example 3 3x + 4y = -10 5x – 2y = 18 There are no opposite like terms. However, if I multiply the second equation by 2, the y’s will be opposites. (The first equation will stay the same.) 10x – 4y = 36

Example 3 Now add the like terms. 13x = 26 Solve. x = 2 Replace x in the first equation with 2 to find y. 3(2) + 4y = -10 6 + 4y = -10 Subtract 6 from both sides.

Example 3 4y = -16 Divide both sides by 4 y = -4 Check by replacing x with 2 and y with –4 in the second equation. 5(2) – 2(-4) = 18 10 + 8 = 18. This is true. The solution is (2, -4)

Example 4 7x – 12y = -22 5x – 8y = -14 I choose to get rid of the x’s. So I will multiply the top equation by 5 and the bottom equation by -7 35x – 60y = -110 -35x + 56y = 98 Now add the equations.

Example 4 -4y = -12 y = 3 That is the second member of the ordered pair. Now find x by replacing y in one of the equations with 3 and solve for x. 7x – 12(3) = -22 7x – 36 = -22 7x = 14 x = 2 (2, 3)

Try these… 2x + 7y = 31 5x – 7y = -45 x – 6y = 2 6x + 6y = 12 (-2, 5) (2, 0) (2, 6) (8, 2)