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We can apply the fact that the sum of opposites is zero to systems of equations so that one of the variables can be eliminated. Look at the following system of equations. What do you observe? x y6 x y2 Zero pair In each equation, there is a “y”. However, the signs in the two equations are opposite. If added together, the “y” and the “-y” would make a zero pair. Add the following two equations together by adding like terms together. x y6 x y2 2x 0 8 Now solve for “x”. 2x = 8 x=4 Now pick one of the two equations and substitute the value of “x” into that equation and solve for “y”. x+y=6 4+y=6 -4 -4 y=2 Solution: (4,2) x y6 x y2 Solution: (4,2) To check your work, substitute the values for the variables into each equation and determine if it is true. x y6 x y2 4 2 6 66 4 2 2 2 2 The solution is correct. Solve the system 4x + 3y = 16 2x – 3y = 8 6x = 24 6 6 x = 4 4x + 3y = 16 2x – 3y = 8 4x + 3y = 16 4(4) + 3y = 16 16 + 3y = 16 -16 -16 3y = 0 3 3 y=0 Solution is (4, 0) Solve the system 2x + 8y = -30 -2x – 10y = 34 2x + 8y = -30 -2x – 10y = 34 -2y = 4 -2 -2 y = -2 2x + 8y = -30 2x + 8(-2) = -30 2x + -16 = -30 + 16 +16 2x = -14 2 2 x = -7 Solution is (-7, -2) Addition worked when the signs of one of the variables were opposite. However, you may encounter a system of equations in which the signs of the variables are the same. In this case, instead of adding, you will be subtracting one equation from the other. Remember that subtraction is really the same as adding the opposite. 4x y 9 3x y 6 The signs of the variables are all positive. Therefore, in order to solve this system, we can subtract the bottom equation from the top equation. Subtract the bottom equation from the top equation. 4x y 9 (-) 3x y 6 x03 or Since we have the value for “x”. Pick one of the equations and substitute the value 3 for “x” and solve for “y”. 3x + y = 6 (3)(3) + y = 6 9+y=6 -9 x3 Solution: -9 y = -3 (3,-3) 4x y 9 3x y 6 Solution: (3,-3) To check your work, substitute the values for the variables into each equation and determine if it is true. 4x y 9 4 ( 3) ( 3) 9 12 3 9 9 9 3x y 6 3( 3) ( 3) 6 9 3 6 66 The solution is correct. Solve the system 3x + 5y = 18 3x + 2y = 9 3x + 5y = 18 -( 3x + 2y = 9) 3y = 9 3 3 y= 3 3x + 5 y = 18 3x + 5(3) = 18 3x + 15 = 18 -15 -15 3x = 3 3 3 x = 1 The solution is (1, 3) Summary of Steps Step 1: The equations must be Standard Form. Standard Form: Ax + By = C Step 2: Determine which variable to eliminate. Look for variables that have the same or opposite coefficients. Step 3: Add or subtract the equations. Solve for the Variable. Step 4: Plug back in to find the other variable. Find the value of the second variable. Step 5: Check your solution. Substitute your ordered pair into BOTH equations. Solve each system of equations by using addition or subtraction. 1. 2x 4 y 18 x 4y 3 2. 3 p 2r 5 3 p 6r 15 3. The sum of two numbers is 85. The difference between the two numbers is 19. What are the two numbers? 2x 4 y 18 x 4y 3 Add the two equations together. 2x 4 y 18 (+) x 4y 3 3x 0 21 Solve for “x”. 3x = 21 3 3 x=7 Solve for “y” by substituting the value for “x” into one of the equations. x 4y 3 7 4y 3 7 7 4 y 4 y1 Solution: (7,1) 3 p 2r 5 3 p 6r 15 Subtract the bottom equation from the top equation. 3 p 2r 5 (−) 3 p 6r 15 4 r 20 Solve for “s” by substituting the value for “r” into one of the equations. 3 p 2r 5 3 p 2(5) 5 3 p 10 5 10 10 3 p 15 Solve for “r”. 3 p 15 3 3 4 r 20 4 4 r 5 p5 Solution: (5,-5) The sum of two numbers is 85. The difference between the two numbers is 19. What are the two numbers? x y 85 x y 19 2x 0 104 2x 104 2 2 x 52 Write an equation for the sum of the numbers. Write an equation for the difference of the numbers. Add the equations together. Solve for x. Next substitute x = 52 into one of the equations. x y 85 x y 19 52 y 85 y 33 The two numbers are 52 and 33. Solve each system of equations by using addition or subtraction. (4) x + 2y = 7 3x – 2y = -3 (5) 2x + 7y = 1 2x + 3y = 9 (6) Twice a number minus a second number is 15. The sum of the two numbers is -6. What are the two numbers?