Solving Systems of Equations: Elimination Method Mr. Pearson Inman Middle School February 2, 2012 - ACCELERATED

BY GRAPHING Y = 2X + 1 Y = -X + 4 (1,3) IS THE SOLUTION   (1,3) IS THE SOLUTION GPS STANDARD: M8A5.b Solving Systems Graphically and Algebraically.

Graphing is not the only way to solve a system of equations Graphing is not the only way to solve a system of equations. We already know a second method called SUBSTITUTION. We have yet another way to solve. This method is called the ELIMINATION Method. You solve by eliminating one variable.

Solve: by ELIMINATION x + y = 12 -x + 3y = -8 Like variables must be lined under each other. Solve: by ELIMINATION x + y = 12 -x + 3y = -8 4y = 4 We need to eliminate (get rid of) a variable. The x’s will be the easiest. So, we will add the two equations. Divide by 4 y = 1 THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. X +Y = 12 x + 1 = 12 -1 -1 x = 11 (11,1) Answer Now check our answers in both equations------

X + Y =12 11 + 1 = 12 12 = 12 -x + 3y = -8 -11 + 3(1) = -8 -11 + 3 = -8 -8 = -8

Solve: by ELIMINATION 5x - 4y = -21 -2x + 4y = 18 Like variables must be lined under each other. Solve: by ELIMINATION 5x - 4y = -21 -2x + 4y = 18 3x = -3 We need to eliminate (get rid of) a variable. The y’s be will the easiest.So, we will add the two equations. Divide by 3 x = -1 THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. 5X - 4Y = -21 5(-1) – 4y = -21 -5 – 4y = -21 5 5 -4y = -16 y = 4 (-1, 4) Answer Now check our answers in both equations------

5x - 4y = -21 5(-1) – 4(4) = -21 -5 - 16 = -21 -21 = -21 -2(-1) + 4(4) = 18 2 + 16 = 18 18 = 18

Solve: by ELIMINATION 2x + 7y = 31 5x - 7y = - 45 Like variables must be lined under each other. Solve: by ELIMINATION 2x + 7y = 31 5x - 7y = - 45 We need to eliminate (get rid of) a variable. The y’s will be the easiest. So, we will add the two equations. Divide by 7 7x = -14 x = -2 THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. 2X + 7Y = 31 2(-2) + 7y = 31 -4 + 7y = 31 4 4 7y = 35 y = 5 (-2, 5) Answer Now check our answers in both equations------

2x + 7y = 31 2(-2) + 7(5) = 31 -4 + 35 = 31 31 = 31 5x – 7y = - 45 5(-2) - 7(5) = - 45 -10 - 35 = - 45 - 45 =- 45

Solve: by ELIMINATION x + y = 30 x + 7y = 6 Like variables must be lined under each other. Solve: by ELIMINATION x + y = 30 x + 7y = 6 We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If one of the x’s was negative, it would be eliminated when we add. So we will multiply one equation by a – 1.

X + Y = 30 X + Y = 30 -X – 7Y = - 6 ( X + 7Y = 6 ) -1 -6Y = 24 - 6 - 6 Now add the two equations and solve. - 6 - 6 Y = - 4 THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. X + Y = 30 X + - 4 = 30 4 4 X = 34 (34, - 4) Answer Now check our answers in both equations------

x + y = 30 34 + - 4 = 30 30 = 30 x + 7y = 6 34 + 7(- 4) = 6 34 - 28 = 6 6 = 6

Solve: by ELIMINATION x + y = 4 2x + 3y = 9 Like variables must be lined under each other. Solve: by ELIMINATION x + y = 4 2x + 3y = 9 We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.

( ) -2 Y = 1 THEN---- X + Y = 4 -2X - 2 Y = - 8 2X + 3Y = 9 Now add the two equations and solve. THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. X + Y = 4 X + 1 = 4 - 1 -1 X = 3 (3,1) Answer Now check our answers in both equations------

x + y = 4 3 + 1 = 4 4 = 4 2x + 3y = 9 2(3) + 3(1) = 9 6 + 3 = 9 9 = 9