Title: Solving Systems of Three Linear Equations in Three Variables
1Solving Systems of Three Linear Equations in
Three Variables
SPI 3103.3.8 Solve systems of three linear
equations in three variables.
2Solutions 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.
3Is (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.
4Methods Used to Solve Systems in 3 Variables
1. Substitution 2. Elimination 3. Cramers
Rule 4. Gauss-Jordan Method .. And others
5Why 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.
6This 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).
18It 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)