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Warm Up 11-20 Simplify the following expressions: 1. 3(3π₯ β 5π¦) 2. -(π¦ + 3π₯) 3. -6(2π₯ + 6π¦) Solve the System of Equations: 1 4. y = 2π₯ β 3 5. y = β π₯ β 2 3 2y β 3π₯ = β4 y=π₯+2 Unit 5 Day 3 Solving Systems by Linear Combinations Essential Questions: How can we solve systems using the linear combinations method? Vocabulary β’ Opposite Coefficients: two terms that have the same numerical coefficient but with opposite signs β’ Example: 2x and -2x Solving Systems Using Linear Combinations 1. Make sure your equations are lined up. 2. If necessary, multiply one or both equations by a number to get opposite coefficients. 3. Combine the equations together (one variable should be eliminated) and solve for the remaining variable. 4. Plug your answer into one of your original equations to solve for the other variable. Example 1: Use linear combinations to solve the linear system. -x + 2y = -8 x + 6y = -16 + 8y = -24 x + 6y = -16 x + 6(-3) = -16 x -18 = -16 + 18 + 18 x=2 Solution: (2, -3) 8 8 y = -3 Check! -x + 2y = -8 -(2) + 2(-3) = -8 -2 - 6 = -8 -8 = -8 x + 6y = -16 2 + 6(-3) = -16 2 - 18 = -16 -16 = -16 Example 2: Use linear combinations to solve the linear system. 3x + 2y = 8 + -5x - 2y = 12 -2x = 20 3x + 2y = 8 3(-10) + 2y = 8 -30 + 2y = 8 + 30 + 30 -2 -2 x = -10 Solution: (-10, 19) 2y = 38 2 2 y = 19 Don't forget to check your solution in both original equations!! Example 3: Use linear combinations to solve the linear system. 2(9x - 3y = 20 ) 3x + 6y = 2 3x + 6y = 2 3(2) + 6y = 2 6 + 6y = 2 -6 -6 18x - 6y = 40 + 3x + 6y = 2 21x = 42 21 x=2 6y = -4 6 6 2 y=- 3 21 2 Solution: (2, ) 3 Don't forget to check your solution in both original equations!! Example 4: Use linear combinations to solve the linear system. 3x + 2y = 46 -3( x + 5y = 11 ) 3x + 2y = 46 3x + 2(-1) = 46 3x - 2 = 46 + 2 + 2 + 3x = 48 3 3 x = 16 3x + 2y = 46 + -3x - 15y = -33 -13y = 13 -13 -13 y = -1 Solution: (16, -1) Don't forget to check your solution in both original equations!! Example 5: Use linear combinations to solve the linear system. -3( 2x - 3y = 0 ) 2( 3x - 2y = 5 ) -6x + 9y = 0 + 6x - 4y = 10 5y = 10 2x - 3y = 0 2x - 3(2) = 0 2x - 6 = 0 +6 +6 2x = 6 x=3 5 5 y=2 Solution: (3, 2) Don't forget to check your solution in both original equations!! Example 6: Use linear combinations to solve the linear system. -1( 2x + y = 5 ) -2x - y = -5 2x + y = 1 + 2x + y = 1 0 = -4 No Solution! Example 7: Use linear combinations to solve the linear system. -2( -2x + y = 3 ) 4x- 2y = -6 -4x + 2y = 6 + -4x + 2y = 6 0=0 All Real Numbers! Summary Essential Questions: How can we solve systems using the linear combinations method? Take one minute to write two sentences answering the essential questions.