1. Solving Systems of Three Linear Equations in Three Variables The Elimination Method SPI 3103.3.8 Solve systems of three linear equations in three variables.

2. Solutions of a system with 3 equations The solution to a system of three linear equations in three variables is an ordered triple. (x, y, z) The solution must be a solution of all 3 equations.

3. Is (–3, 2, 4) a solution of this system? 3x + 2y + 4z = 11 2x – y + 3z = 4 5x – 3y + 5z = –1 3(–3) + 2(2) + 4(4) = 11 2 (–3) – 2 + 3 (4) = 4 5 (–3) – 3 (2) + 5 (4) = –1    Yes, it is a solution to the system because it is a solution to all 3 equations.

4. Methods Used to Solve Systems in 3 Variables 1. Substitution 2. Elimination 3. Cramer’s Rule 4. Gauss-Jordan Method ….. And others

5. Why not graphing? While graphing may technically be used as a means to solve a system of three linear equations in three variables, it is very tedious and very difficult to find an accurate solution. The graph of a linear equation in three variables is a plane.

6. This lesson will focus on the Elimination Method.

7. Use elimination to solve the following system of equations. x – 3y + 6z = 21 3x + 2y – 5z = –30 2x – 5y + 2z = –6

8. Step 1 Rewrite the system as two smaller systems, each containing two of the three equations.

9. x – 3y + 6z = 21 3x + 2y – 5z = –30 2x – 5y + 2z = –6 x – 3y + 6z = 21 3x + 2y – 5z = –30 2x – 5y + 2z = –6

10. Step 2 Eliminate THE SAME variable in each of the two smaller systems. Any variable will work, but sometimes one may be a bit easier to eliminate. I choose x for this system.

11. (x – 3y + 6z = 21) 3x + 2y – 5z = –30 –3x + 9y – 18z = –63 3x + 2y – 5z = –30 11y – 23z = –93 (x – 3y + 6z = 21) 2x – 5y + 2z = –6 –2x + 6y – 12z = –42 2x – 5y + 2z = –6 y – 10z = –48 (–3) (–2)

12. Step 3 Write the resulting equations in two variables together as a system of equations. Solve the system for the two remaining variables.

13. 11y – 23z = –93 y – 10z = –48 11y – 23z = –93 –11y + 110z = 528 87z = 435 z = 5 y – 10(5) = –48 y – 50 = –48 y = 2 (–11)

14. Step 4 Substitute the value of the variables from the system of two equations in one of the ORIGINAL equations with three variables.

15. x – 3y + 6z = 21 3x + 2y – 5z = –30 2x – 5y + 2z = –6 I choose the first equation. x – 3(2) + 6(5) = 21 x – 6 + 30 = 21 x + 24 = 21 x = –3

16. Step 5 CHECK the solution in ALL 3 of the original equations. Write the solution as an ordered triple.

17. x – 3y + 6z = 21 3x + 2y – 5z = –30 2x – 5y + 2z = –6 –3 – 3(2) + 6(5) = 21 3(–3) + 2(2) – 5(5) = –30 2(–3) – 5(2) + 2(5) = –6    The solution is (–3, 2, 5).

18. It is very helpful to neatly organize your work on your paper in the following manner. (x, y, z)

19. Try this one. x – 6y – 2z = –8 –x + 5y + 3z = 2 3x – 2y – 4z = 18 (4, 3, –3)

20. Here’s another one to try. –5x + 3y + z = –15 10x + 2y + 8z = 18 15x + 5y + 7z = 9 (1, –4, 2)