Solving Systems of
Equations with Three
Variables
By: Janeat Toma
Describing Solutions to a System of Three Equations in
Three Variables
● Ax+By+Cz=D
● Each equation defines a flat plane that can be graphed on a 3D x-y-z
graph.
● The solution is when these three planes cross a single point.
● Another type of solution has an infinite number of points: a three
dimensional straight line.
● To solve for single point solutions, we can use Elimination or Substitution.
● No solution occurs in some systems such as parallel or triangular planes.
Visualizing Solutions to a System of Three
Equations in Three Variables
Solving a System of Equations Algebraically
x-3y+3z=-4
2x+3y-z=15
4x-3y-z=19
Solution:
Example:
To solve a system of three equations in
three variables, we will be using the
linear combination method. This time
we will take two equations at a time to
eliminate one variable and using the
resulting equations in two variables to
eliminate a second variable and solve
for the third.
Identifying Inconsistent Systems and Dependent
Equations
● When the equations in a system of two
equations with two variables are
dependent, the system has infinitely
many solutions
○ This is NOT always true for systems of three
equations with three variables.
■ A system can have dependent
equations and still be inconsistent in
this case.
● The illustration demonstrates the
different possibilities
Using a Matrix to Solve a System of Equations
● Step 1: Write the coefficients in a matrix using a vertical line to represent
equals signs.
● Step 2: Find the inverse of the matrix that’s left of the equals signs.
● Step 3: Multiply the inverse matrix by the part of the matrix that is right of the
equals sign.
Example of Using a Matrix
x + y - z = -2
2x - y + z = 5
-x + 2y + 2z = 1
[
1 1 -1
2 -1 1
-1 2 2
-2
5
1
[
R1 + R3
[
1 1 -1
2 -1 1
0 3 1
-2
5
-1
[
Example of Using a Matrix Continued
-2R1+ R2
[
1 1 -1
0 -3 3
0 3 1
-2
9
-1
[
[
[
R2 + R3
1 1 -1
0 -3 3
0 0 4
-2
9
8
From the third row, 4z = 8.
To solve for z, divide both sides by 4, z= 2.
From the second row, -3y + 3z = 9. Substitute z=2, -3y + 6 = 9. Subtract 6 on both sides= -3y = 3.
Divide -3 on both sides, y = -1.
From the first row, x + y - z = -2. Solving for x, substitue y= -1 and z= 2.
x - 1 - 2 = -2.
x -3 = -2. Add three to both sides, x= 1.
The solution is (1, -1, 2).
