Solving Systems of Three Linear Equations in

Three Variables

- The Elimination Method

Solve systems of three linear equations in

three variables.

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.

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

P

P

P

Yes, it is a solution to the system because it

is a solution to all 3 equations.

Methods Used to Solve Systems in 3 Variables

1. Substitution 2. Elimination 3. Cramers

Rule 4. Gauss-Jordan Method .. And others

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.

This lesson will focus on the Elimination

Method.

Use elimination to solve the following system of

equations. x 3y 6z 21 3x 2y 5z

30 2x 5y 2z 6

Step 1 Rewrite the system as two smaller

systems, each containing two of the three

equations.

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

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.

(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)

Step 3 Write the resulting equations in two

variables together as a system of

equations. Solve the system for the two

remaining variables.

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)

Step 4 Substitute the value of the variables

from the system of two equations in one of the

ORIGINAL equations with three variables.

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

Step 5 CHECK the solution in ALL 3 of the

original equations. Write the solution as an

ordered triple.

P

3 3(2) 6(5) 21 3(3) 2(2) 5(5)

30 2(3) 5(2) 2(5) 6

x 3y 6z 21 3x 2y 5z 30 2x 5y 2z

6

P

P

The solution is (3, 2, 5).

It is very helpful to neatly organize your work

on your paper in the following manner.

(x, y, z)

Try this one. x 6y 2z 8 x 5y 3z

2 3x 2y 4z 18 (4, 3, 3)

Heres another one to try. 5x 3y z

15 10x 2y 8z 18 15x 5y 7z 9

(1, 4, 2)