The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.

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)