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Lesson 7.4A Solving Linear Systems Using Elimination

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Keys to Know Solving with Elimination: When you combine the equations to get rid of (eliminate) one of the variables. Possible solutions are: One Solution No Solutions Infinite Solutions

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Steps for Using Elimination 1) Write both equations in standard form (Ax + By = C) so that variables and = line up 2) Multiply one or both equations by a number to make opposite coefficients on one variable. 3) Add equations together (one variable should cancel out) 4) Solve for remaining variable. 5) Substitute the solution back in to find other variable. 6) Write the solution as an ordered pair 7) Check your answer

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Example 1: 5x + y = 12 5x + y = 12 3x – y = 4 3x – y = 4 8x = 16 8x = 16 8 8 8 8 x = 2 x = 2 5(2) + y = 12 10 + y = 12 y = 2 The solution is: (2, 2) Step 1: Put both equations in standard form. Step 2: Check for opposite coefficients. Step 3: Add equations together Step 4: Solve for x Step 5: Substitute 2 in for x to solve for y (in either equation) Already Done y and –y are already opposites

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Your Turn Ex. 2 2x + y = 0 Ex. 2 2x + y = 0 -2x + 3y = 8 -2x + 3y = 8 Answer: (-1, 2)

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Example 3 3x + 5y = 10 3x + 5y = 10 3x + y = 2 3x + y = 2 3x + 5y = 10 3x + 5y = 10 -1(3x + y) = -1(2) 4y = 8 4y = 8 y = 2 y = 2 Now plug (2) in for y. 3x + 2 = 2 X = 0 Solution is : (0,2) When you add these neither variable drops out SO…. We need to change 1 or both equations by multiplying the equation by a number that will create opposite coefficients. When we need to create opposite coefficients 3x + 5y = 10 -3x – y = -2 -3x – y = -2 Multiply the bottom equation by negative one to eliminate the x

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4) -2x + 3y = 6 x – 4y = -8 x – 4y = -8 -2x + 3y= 6 -2x + 3y = 6 2( x – 4y) = -8(2) 2x - 8y = -16 -5 y = -10 -5 y = -10 y = 2 y = 2 Now plug (2) in for y into any of the 4 equations. -2x + 3(2) = 6 -2x + 6 = 6 -2x = 0 x = 0 x = 0 Solution is: (0, 2)Check your work! We will need to change both equations. We will have the y value drop out.

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Your Turn Ex. 55x – 2y = 12 2x – 2y = -6 Ex. 6-3x + 6y = 9 x - 2y = -3 x - 2y = -3 Ex. 72x + 4y = 8 x + 2y = 3 x + 2y = 3 (6, 9) 0=0 Infinite solutions 0=2 No Solutions

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