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**Solving Systems of Equations**

Section 4.2

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**Useful when all variable have coefficients other than 1.**

Substitution Method (Windshield Wipers) Linear Combinations (Elimination) Useful technique for solving systems in which a variable has a coefficient of 1. Useful when all variable have coefficients other than 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. 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 additive inverses. 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.

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**Substitution Method (1, 8) y = 3x + 5 2x + 4y = 34 y = 3x + 5**

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**Substitution Method (1, ) x – 4y = -1 2x + 2y = 3 x - 4y = -1**

2(4y-1) + 2y = 3 8y – 2 + 2y = 3 10y – 2 = 3 x = 10 y = 5 x = 2 - 1 y = x = 1 (1, )

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**Decide which variable you want to eliminate.**

Linear Combination 3x – 5y = 14 2x + 4y = -20 I think I’ll choose to eliminate the y variable. Decide which variable you want to eliminate.

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**Linear Combination 4 5 (-2, -4) 3x – 5y = 14 2x + 4y = -20**

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**Decide which variable you want to eliminate.**

Linear Combination 2x + 7y = 48 3x + 5y = 28 I think I’ll choose to eliminate the x variable. Decide which variable you want to eliminate.

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**Linear Combination 3 -2 (-4, 8) 2x + 7y = 48 3x + 5y = 28**

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**Decide which variable you want to eliminate.**

Linear Combination 4x + 3y = -19 6x + 5y = -32 I think I’ll choose to eliminate the x variable. Decide which variable you want to eliminate.

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**Linear Combination 3 -2 4x + 3y = -19 6x + 5y = -32 12x + 9y = -57**

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**Substitution Method Parallel Lines y = -2x - 6 6x + 3y = 11**

- 18 = 11 Parallel Lines

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**Infinitely many solutions**

Substitution Method x = 5y + 1 2x y = 2 2(5y + 1) y = 2 10y y = 2 2 = 2 Infinitely many solutions same lines

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