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Substitution Method (Windshield Wipers) Useful technique for solving systems in which a variable has a coefficient of 1. Step 1: Solve one of the equations for either one of its variables. Step 2: Substitute the expression you have for Step 1 into the other equation and solve for the remaining variable. Step 3: Substitute the value from Step 2 back into the equation from Step 1 and solve for the second variable. Step 4 : Check your solution in both of the original equations. Linear Combinations (Elimination) Useful when all variables have coefficients other than 1. Step 1: Arrange both equations so the like terms line up in same column. Step 2: Multiply one or both of the equations by the same number so the coefficients of one of the variables are opposites. Step 3: Add the equations together. One of the variables should eliminate because the coefficients will add to zero. Step 4: Solve for the remaining variable. Step 5: Substitute the solution from Step 4 Into either of the original equations and solve for the other variable. Step 6 : Check your solution in both of the original equations. y = 3x + 5 2x + 4y = 34 y = 3x + 5 2x + 4y = 34 2x + 4(3x + 5) = 34 2x + 12x + 20 = 34 y = 3(1) + 5 y=3+5 y=8 14x + 20 = 34 14x = 14 x = 1 x – 4y = -1 2x + 2y = 3 x - 4y = -1 2x + 2y = 3 x = 4y - 1 2(4y-1) + 2y = 3 8y – 2 + 2y = 3 x= 1 4 2 x=2-1 x=1 1 10y – 2 = 3 10 y = 5 1 y = 2 1 2 3x – 5y = 14 2x + 4y = -20 Decide which variable you want to eliminate. I think I’ll choose to eliminate the y variable. 3x – 5y = 14 2x + 4y = -20 56 3(-2) – 5y = 14 10x + 20y = -100 -6 - 5y = 14 12x - 20y = 22x = -44 x = -2 -5y = 20 y = -4 2x + 7y = 48 3x + 5y = 28 Decide which variable you want to eliminate. I think I’ll choose to eliminate the x variable. 2x + 7y = 48 3x + 5y = 28 6x + 21y = 144 3x + 5(8) = 28 -6x - 10y = -56 3x + 40 = 28 11y = 88 y = 8 3x = -12 x = -4 4x + 3y = -19 6x + 5y = -32 Decide which variable you want to eliminate. I think I’ll choose to eliminate the x variable. 4x + 3y = -19 6x + 5y = -32 12x + 9y = -57 6x + 5(-7) = -32 -12x - 10y = 64 6x - 35 = -32 6x = 3 -y = 7 y = -7 x 1 2 , 7 1 2 y = -2x - 6 6x + 3y = 11 6x + 3(-2x - 6) = 11 6x - 6x - 18 = 11 - 18 = 11 x = 5y + 1 2x - 10y = 2 2(5y + 1) - 10y = 2 10y + 2 - 10y = 2 2= 2