APS

1. Simultaneous Equations with 3
unknowns

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 3(–3) + 2(2) + 4(4) = 11 P
2x – y + 3z = 4 2(–3) – 2 + 3(4) = 4 P
5x – 3y + 5z = –1 5(–3) – 3(2) + 5(4) = –1P
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 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) (–3) (x – 3y + 6z = 21) (–2)
3x + 2y – 5z = –30 2x – 5y + 2z = –6
–3x + 9y – 18z = –63 –2x + 6y – 12z = –42
3x + 2y – 5z = –30 2x – 5y + 2z = –6
11y – 23z = –93 y – 10z = –48

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

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 P
–3 – 3(2) + 6(5) = 21
3x + 2y – 5z = –30 P
3(–3) + 2(2) – 5(5) = –30
2x – 5y + 2z = –6 2(–3) – 5(2) + 2(5) = –6
P
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)