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Transcript
5-4 Elimination Using Multiplication aka Linear Combination Algebra 1 Glencoe McGraw-Hill Linda Stamper Often the equations are not ready for one variable to cancel. You will need to create the opposites. 3x 8y 13 4x 4 y 12 Multiply one or both equations by a number to obtain coefficients that are opposites for one of the variables. 33x 8y 13 4x 4 y 12 24x 4 y 122 Choose one of the variables to create an opposite. 3x 8y 13 8x 8y 24 11 11x x 1 Multiply by a number needed to make one variable opposites. Add your equation because you now have opposites. Substitute the solved value into either of the original equations to find the value for the other variable. Yeah a handout. I Write solutiondoas an ordered pair. not have to copy these notes in my notebook! 3 1 8y 13 3 8y 13 8y 16 y 2 (–1,2) 33x 8y 13 4x 4 y 12 43x 8y 134 34x 4 y 12 3 12x 32y 52 12x 12y 36 44 y 88 While you can choose either of the variables to make opposites, choosing wisely may save you some work. Here is the work if the x variable is used to create the opposite. y 2 3 x 82 13 3x 16 13 16 16 3x 3 x 1 (-1,2) Solve the linear system using elimination. Example 1 Example 2 3x y 8 7 x 3y 8 3x 4 y 6 2x 5y 19 Example 1 Solve the linear system. 3 3xx yy 88 7 x 3y 8 33x y 83 9x 3y 24 7x 3y 16x 8 32 x 2 3 2 y 8 6 y 8 y 2 (2,2) Example 2 Solve the linear system. 53x 4 y 65 33x 4 y 6 2x 5y 19 42x 5y 19 4 15x 20y 30 8x 20y 76 23x 46 x 2 3 2 4 y 6 6 4y 6 4 y 12 y 3 (–2,3) Example 2 Solve the linear system. 23x 4 y 62 33x 4 y 6 2x 5y 19 32x 5y 19 3 6x 8y 12 6x 15y 57 23y 69 y 3 3 x 43 6 3x 12 6 3x 6 x 2 (–2,3) Solving A Linear System By Elimination 1) Arrange the equations with like terms in columns. 2) Multiply, if necessary, one or both equations by the number needed to make one of the variables an opposite. 3) Add the equations when one of the variables have opposites. Then solve. 4) Substitute the value solved into either of the original equations and solve for the other variable. 5) Check the ordered pair solution in each of the original equations. When solving a system by elimination, rearrange the terms so that the corresponding variables are vertically stacked. 3x 6y 12 x 3yy 66 3x 6y 12 3x 6y 12 x 3y 6 3 x 3y 6(3) Substitute the solved value into either of the original equations. x 3(2) 6 Write answer as an ordered pair. x66 x 0 x0 3x 9y 18 15 y 30 y 2 (0,2) Solve the linear system using elimination. Example 3 Example 4 3x 2y 8 2y 12 5x 13 4x 3y 5x 2y 1 Example 5 3y 5x 15 6x 2y 18 Example 6 Write a linear system and then solve. Five times the first number minus three times the second number is six. Two times the first number minus five times the second number is ten. Find the numbers. Assign labels. Translate each sentence. Solve the system. Write a sentence to give the answer. Example 3 Solve the linear system. 33x 2y 8 3x 2y 8 13x 2y 8 1 2y 12 5x 5x 2y 12 Rewrite in standard form. 3 (2) 2y 8 6 2y 8 2 2y 1y (2,1) 3x 2y 8 5x 2y 12 2x 4 x 2 Example 4 Solve the linear system. 13 4 4 x 3y 5x 2y 1 4x 3y 13 5x 2y 1 Rewrite in standard form. 13 4 1 3y 13 4 3y 9 3y 3y (–1,3) 8x 6y 26 15x 6y 3 23x 23 x 1 Example 5 Solve the linear system. 3y 5x 15 6x 2y 18 5x 3y 15 6x 2y 18 3y 5 3 15 3y 15 15 3y 0 0 3 y0 3,0 y 10x 6y 30 18x 6y 54 8x 24 x 3 Example 6 Five times the first number minus 3 times the second number is six. Two times the first number minus five times the second number is 10. Find the numbers. 5x 3y 6 10x 6y 12 2x 5y 10 10x 25y 50 19 y 38 Translate each sentence. y 2 Solve the system. Let x = first number Let y = second number Write a sentence to give the answer. The numbers are -2 and 0. 5x 3 2 6 5x 6 6 5x 0 x0 Practice Problems 1) 2x 3y 4 4x 5y 8 1) (2,0) 2) 6x 2y 2 3x 3y 9 2) (1,–2) 3) 5x 4 y 3 2x 8y 2 1 1 3) , 3 3 4) Six times the first number plus two times the second number is two. Four times the first number plus three times the second number is eight. Find the numbers. The numbers are -1 and 4. 5-A6 Page 276-278 #7–17,30,34-36,44-48.