Solving Systems of Three Linear Equations in Three Variables

1
Solving Systems of Three Linear Equations in
Three Variables

• The Elimination Method

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
P
P
P
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. Cramers
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) 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

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).
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

Heres another one to try. 5x 3y z
15 10x 2y 8z 18 15x 5y 7z 9
(1, 4, 2)