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Systems of Equations Solve by Elimination Solve by Elimination. . . . System of equations: x + y = 12 -x + 3y = -8. 1. Both equations have to be in Standard Form—the x’s and the y’s on one side of the equation and the number by itself on the other side. 2. The like terms have to be lined up. System of equations: x + y = 12 -x + 3y = -8. We can eliminate one of the variables— x or y by adding the two equations together. Add the two equations together. What do you get? Solve by Elimination. . . . x + y = 12 -x + 3y = -8 4y = 4 4 4 y= 1 y=1 What is x? Substitute what we know for y. y = 1 into one of our original equations. x + y = 12 x + 1 = 12 Solve for x. x + 1 = 12 -1 -1 x = 11 y = 1 and x = 11 The solution to the system of equations: x + y = 12 -x + 3y = -8 is (11, 1). Now check your answer by substituting into the original equations. . . . (11, 1) Remember it is (x, y). x + y = 12 -x + 3y = -8 11 + 1 = 12 -11 + 3(1) = -8 Solve by Elimination. . . . System of equations: 5x - 4y = -21 -2x + 4y = 18. 1. Both equations have to be in Standard Form—the x’s and the y’s on one side of the equation and the number by itself on the other side. 2. The like terms have to be lined up. System of equations: 5x - 4y = -21 -2x + 4y = 18. We can eliminate one of the variables— x or y by adding the two equations together. Add the two equations together. What do you get? Solve by Elimination. . . . 5x - 4y = -21 -2x + 4y = 18 3x = -3 3 3 x = -1 x = -1 What is y? Substitute what we know for x, x = -1, into one of our original equations. 5x - 4y = -21 5(-1) + 4y = -21 Solve for y. 5(-1) - 4y = -21 -5 - 4y = -21 +5 +5 - 4y = -16 -4 -4 y=4 x = -1 and y = 4 The solution to the system of equations: 5x - 4y = -21 -2x + 4y = 18 is (-1, 4). Now check your answer by substituting into the original equations. . . . (-1, 4) Remember it is (x, y). 5x - 4y = -21 -2x + 4y = 18 5(-1) – 4(4) = -21 -2(-1) + 4(4) = 18 Solve by Elimination. . . . System of equations: 2x + 7y = 31 5x - 7y = -45. 1. 2. Both equations have to be in Standard Form—the x’s and the y’s on one side of the equation and the number by itself on the other side. The like terms have to be lined up. System of equations: 2x + 7y = 31 5x - 7y = -45 We can eliminate one of the variables— x or y by adding the two equations together. Add the two equations together. What do you get? Solve by Elimination. . . . 2x + 7y = 31 5x - 7y = -45 7x = -14 7 7 x = -2 x = -2 What is y? Substitute what we know for x, x = -2, into one of our original equations. 2x + 7y = 31 2(-2) + 7y = 31 Solve for y. 2(-2) + 7y = 31 -4 + 7y = 31 +4 +4 7y = 35 7 7 y=5 x = -2 and y = 5 The solution to the system of equations: 2x + 7y = 31 5x - 7y = -45 is (-2, 5). Now check your answer by substituting into the original equations. . . . (-2, 5) Remember it is (x, y). 2x + 7y = 31 5x - 7y = -45 2(-2) + 7(5) = 31 5(-2) - 7(5) = -45 Solve by Elimination. . . . System of equations: x + y = 30 x + 7y = 6. 1. Both equations have to be in Standard Form—the x’s and the y’s on one side of the equation and the number by itself on the other side. 2. The like terms have to be lined up. System of equations: x + y = 30 x + 7y = 6 We cannot simply add them together because that does not eliminate one of the variables. We have to multiply one of the equations by -1 and them add the two equations together. Multiply one of the equations by -1 and then add them together. What do you get? Solve by Elimination. . . . -1(x + y = 30) -x - y = -30 x + 7y = 6 6y = -24 6 6 y = -4 y = -4 What is x? Substitute what we know for y, y = -4, into one of our original equations. x + y = 30 x + -4 = 30 Solve for x. x + -4 = 30 +4 +4 x = 34 x = 34 and y = -4 The solution to the system of equations: x + y = 30 x + 7y = 6 is (34, -4). Now check your answer by substituting into the original equations. . . . (34, -4) Remember it is (x, y). x + y = 30 x + 7y = 6 34 + (-4) = 30 34 + 7(-4) = 6 Solve by Elimination. . . . System of equations: x + y = 4 2x + 3y = 9. 1. Both equations have to be in Standard Form—the x’s and the y’s on one side of the equation and the number by itself on the other side. 2. The like terms have to be lined up. System of equations: x + y = 4 2x + 3y = 9 We cannot simply add them together because that does not eliminate one of the variables. Multiply the first equation by -2 and then add them together. What do you get? Solve by Elimination. . . . -2(x + y = 4) -2x - 2y = -8 2x + 3y = 9 1y = 1 y= 1 y=1 What is x? Substitute what we know for y, y = 1, into one of our original equations. x+y=4 x+1=4 Solve for x. x+1=4 -1 -1 x=3 x = 3 and y = 1 The solution to the system of equations: x+y=4 2x + 3y = 9 is (3, 1). Now check your answer by substituting into the original equations. . . . (3, 1) Remember it is (x, y). x+y=4 2x + 3y = 9 3+1= 4 2(3) + 3(1) = 9 System of equations: 2x + y = 20 6x - 5y = 12 We cannot simply add them together because that does not eliminate one of the variables. Multiply the first equation by 5 and then add them together. What do you get? Solve by Elimination. . . . 5(2x + y = 20) 10x + 5y = 100 6x - 5y = 12 16x = 112 x= 7 x=7 What is y? Substitute what we know for x, x = 7, into one of our original equations. 2x + y = 20 2(7) + y = 20 Solve for x. 14 + y = 20 -14 -14 y=6 x = 7 and y = 6 The solution to the system of equations: 2x + y = 20 6x - 5y = 12 is (7, 6). Now check your answer by substituting into the original equations. . . . (7, 6) Remember it is (x, y). 2x + y = 20 6x - 5y = 12 2(7) + 6 = 20 6(7) - 5(6) = 12 System of equations: -5x – 8y = 17 2x - 7y = -17 We cannot simply add them together because that does not eliminate one of the variables. Multiply the first equation by 2 and the second equation by 5 and then add them together. What do you get? Solve by Elimination. . . . 2(-5x - 8y = 17) 5(2x – 7y = -17) -10x – 16y = 34 10x - 35y = -85 -51y = -51 y= 1 y=1 What is x? Substitute what we know for x, y = 1, into one of our original equations. -5x - 8y = 17 -5x – 8(1) = 17 Solve for x. -5x – 8 = 17 +8 +8 -5x = 25 x = -5 and y = 1 The solution to the system of equations: -5x - 8y = 17 2x - 7y = -17 is (-5, 1). Now check your answer by substituting into the original equations. . . . (-5, 1) Remember it is (x, y). -5x - 8y = 17 2x - 7y = -17 -5(-5) – 8(1) = 17 2(-5) - 7(1) = -17 For Elimination 1. First the two equations need to be in standard form and the like terms added up. 2. Then multiply one or both equations until you get two opposite like terms. 3. Add the two equations. 4. Solve for the first variable. 5. Plug the first variable back in to find the other variable. 6.Plug your answers back into the original equation to solve.