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BY GRAPHING Y = 2X + 1 Y = -X + 4   (1,3) IS THE SOLUTION

Graphing is not the only way to solve a system of equations Graphing is not the only way to solve a system of equations. It is not really the best way because it has to be graphed perfectly and some answers are not integers. SOOOO We need to learn another way!!!!

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 7x = -14 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 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