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Name_______________________________________ Ms. Serzo Date__________________________ Solving Systems of Equations Algebraically (Elimination Method) 1. Solve this system of equations and check: x - 2y = 14 x + 3y = 9 a. First, be sure that the variables are "lined up" under one another. In this problem, they are already "lined up". x - 2y = 14 x + 3y = 9 b. Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be the same or negatives of one another. Looks like "x" is the easier variable to eliminate in this problem since the x's already have the same coefficients. x - 2y = 14 x + 3y = 9 c. Now, in this problem we need to subtract to eliminate the "x" variable. Subtract ALL of the sets of lined up terms. (Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) x - 2y = 14 -x - 3y = - 9 d. Solve this simple equation. - 5y = 5 -5y = 5 y = -1 e. Plug "y = -1" into either of the ORIGINAL equations to get the value for "x". x - 2y = 14 x - 2(-1) = 14 x + 2 = 14 x = 12 f. Check: substitute x = 12 and y = -1 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! x - 2y = 14 12 - 2(-1) = 14 12 + 2 = 14 14 = 14 (check!) x + 3y = 9 12 + 3(-1) = 9 12 - 3 = 9 9 = 9 (check!) 1 2. Solve this system of equations and check: 4x + 3y = -1 5x + 4y = 1 a. You can probably see the dilemma with this problem right away. Neither of the variables have the same (or negative) coefficients to eliminate. Yeek! 4x + 3y = -1 5x + 4y = 1 b. In this type of situation, we must MAKE the coefficients the same (or negatives) by multiplication. You can MAKE either the "x" or the "y" coefficients the same. Pick the easier numbers. In this problem, the "y" variables will be changed to the same coefficient by multiplying the top equation by 4 and the bottom equation by 3. Remember: * you can multiply the two differing coefficients to obtain the new coefficient if you cannot think of another smaller value that will work. * multiply EVERY element in each equation by your adjustment numbers. c. Now, in this problem we need to subtract to eliminate the "y" variable. (Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) d. Plug "x = -7" into either of the ORIGINAL equations to get the value for "y". e. Check: substitute x = -7 and y = 9 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! 4(4x + 3y = -1) 3(5x + 4y = 1) 16x + 12y = -4 15x + 12y = 3 16x + 12y = -4 -15x - 12y = - 3 x =-7 5x + 4y = 1 5(-7) + 4y = 1 -35 + 4y = 1 4y = 36 y=9 4x + 3y = -1 4(-7) +3(9) = -1 -28 + 27 = -1 -1 = -1 (check!) 5x + 4y = 1 5(-7) + 4(9) = 1 -35 + 36 = 1 1 = 1 (check!) 2