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BY GRAPHING Y = 2X + 1 Y = -X + 4 (1,3) IS THE SOLUTION Example #1: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Simplify and solve the equation. 7x = -21 x = -3 y = 4x 3x + y = -21 Step 3: Substitute what you solved into Step 1 y = 4x y = 4(-3) y = -12 Solution to the system is (-3, -12). y = 4x 3x + y = -21 Step 4: Check the solution in both equations. Solution to the system is (-3,-12). y = 4x -12 = 4(-3) -12 = -12 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 Example #2: Solve using “Substitution” x + y = 10 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2 x + y = 10 5x – y = 2 Simplify and solve the equation. 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x=2 x + y = 10 5x – y = 2 Step 3: Substitute back what you solved into step 1 y = -x + 10 y = -(2) + 10 y=8 Solution to the system is (2,8). Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The x’s will be the easiest. So, we will add the two equations. Solve: by ELIMINATION x + y = 12 -x + 3y = -8 Divide by 4 4y = 4 y=1 THEN---- 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 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The y’s be will the easiest.So, we will add the two equations. Solve: by ELIMINATION 5x - 4y = -21 -2x + 4y = 18 Divide by 3 3x = -3 x = -1 THEN---- 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) Now check our answers in both equations-----Answer 5x - 4y = -21 5(-1) – 4(4) = -21 -5 - 16 = -21 -21 = -21 -2x + 4y = 18 -2(-1) + 4(4) = 18 2 + 16 = 18 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The y’s will be the easiest. So, we will add the two equations. Solve: by ELIMINATION 2x + 7y = 31 5x - 7y = - 45 Divide by 7 7x = -14 x = -2 THEN---- 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) Now check our answers in both equations-----Answer 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 Like variables must be lined under each other. 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. Solve: by ELIMINATION x + y = 30 x + 7y = 6 X + Y = 30 X + Y = 30 ( X + 7Y = 6 ) -1 -X – 7Y = - 6 -6Y = 24 Now add the two equations and solve. -6 -6 Y=-4 THEN---- 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) Now check our answers in both equations-----Answer x + y = 30 34 + - 4 = 30 30 = 30 x + 7y = 6 34 + 7(- 4) = 6 34 - 28 = 6 6=6 Like variables must be lined under each other. 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. Solve: by ELIMINATION x+ y=4 2x + 3y = 9 ( X + Y = 4 ) -2 2X + 3Y = 9 -2X - 2 Y = - 8 2X + 3Y = 9 Y=1 Now add the two equations and solve. THEN---- 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) Now check our answers in both equations-----Answer x+y=4 3+1=4 4=4 2x + 3y = 9 2(3) + 3(1) = 9 6+3=9 9=9