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Use elimination to solve the system of equations.

1. 2. 3. Solving a Linear System in Three Variables. Use elimination to solve the system of equations. 5 x – 2 y – 3 z = –7. 2 x – 3 y + z = –16. 3 x + 4 y – 2 z = 7. Step 1 Eliminate one variable. Step 2 Eliminate another variable. Then solve for the remaining variable.

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Use elimination to solve the system of equations.

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  1. 1 2 3 Solving a Linear System in Three Variables Use elimination to solve the system of equations. 5x – 2y – 3z = –7 2x – 3y + z = –16 3x + 4y – 2z = 7 Step 1 Eliminate one variable. Step 2 Eliminate another variable. Then solve for the remaining variable. Step 3 Use one of the equations in your 2-by-2 system to solve for y. Step 4 Substitute for x and y in one of the original equations to solve for z.

  2. Multiply equation - by 3, and add to equation . 2 5 1 2 4 2 3 3 2 3 2 1 Multiply equation - by 2, and add to equation . Use equations and to create a second equation in x and y. Continued 5x – 2y – 3z = –7 5x – 2y – 3z = –7 3(2x –3y + z = –16) 6x – 9y + 3z = –48 11x – 11y = –55 1 3x + 4y – 2z = 7 3x + 4y – 2z = 7 4x – 6y + 2z = –32 2(2x –3y + z = –16) 7x – 2y = –25

  3. 4 5 Continued 11x – 11y = –55 You now have a 2-by-2 system. 7x – 2y = –25

  4. 4 5 4 5 Multiply equation - by –2, and equation - by 11 and add. Continued Step 2 Eliminate another variable. Then solve for the remaining variable. You can eliminate y by using methods from Lesson 3-2. –2(11x – 11y = –55) –22x + 22y = 110 1 11(7x – 2y = –25) 77x – 22y = –275 55x = –165 1 x = –3 Solve for x.

  5. 4 Continued Step 3 Use one of the equations in your 2-by-2 system to solve for y. 11x – 11y = –55 1 11(–3) – 11y = –55 Substitute –3 for x. 1 Solve for y. y = 2

  6. 2 Continued Step 4 Substitute for x and y in one of the original equations to solve for z. 2x – 3y + z = –16 Substitute –3 for x and 2 for y. 1 2(–3) – 3(2) + z = –16 z = –4 Solve for y. 1 The solution is (–3, 2, –4).

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