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3.6 Systems with Three Variables
1. Solving Three-Variable Systems by Elimination
2. Solving Three-Variable Systems by Substitution
1) Solving Three-Variable Systems by Elimination
• A three-variable system produces a 3D graph that is a plane
• A three equation system produces three planes
• The planes may never intersect, intersect once, or have an infinite number of intersections
• Use elimination to solve
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
1
2
3{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
1
2
3
You have to work with what you are given to eliminate one variable at a time.
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
1
2
3
1 2
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
x – 3y + 3z = -4
2x + 3y – z = 15
1
2
3
1 2
1
2
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
x – 3y + 3z = -4
2x + 3y – z = 15
3x + 2z = 11
1
2
3
1 2
1
2
4 New two-variable equation
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
1
2
3
2 3
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
2x + 3y – z = 15
4x – 3y – z = 19
1
2
3
2
2
3
3
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 1: Add and to cancel y.
2x + 3y – z = 15
4x – 3y – z = 19
6x - 2z = 34
1
2
3
2
2
3
5 New two-variable equation
3
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 2: Add and to solve for x.
1
2
3
4 5
{
1) Solving Three-Variable Systems by Elimination
Example 1:Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 154x – 3y – z = 19
Step 2: Add and to solve for x. 3x + 2z = 11 6x - 2z = 34
1
2
3
4
5
4 5
{
1) Solving Three-Variable Systems by Elimination
Example 1:Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 154x – 3y – z = 19
Step 2: Add and to solve for x. 3x + 2z = 11 6x - 2z = 34 9x = 45 x = 5
1
2
3
4
5
4 5
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Step 3: Sub x in to solve for z.
3(5) + 2z = 11
2z = -4
z = -2
1
2
3
4Sub x = 5 into 4
4
{
1) Solving Three-Variable Systems by Elimination
Example 1:Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 154x – 3y – z = 19
Step 4: Sub x and z into one of the original equations. Solve for y.
x – 3y + 3z = -4 5 – 3y + 3(-2) = -4
5 – 3y – 6 = -4 -3y = -4 + 6 – 5 -3y = -3 y = 1
1
2
3
1Sub x = 5, z = -2 in 1
{
1) Solving Three-Variable Systems by Elimination
Example 1:
Solve by elimination. x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
Therefore, the solution is (5, 1, -2).
1
2
3{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
1
2
3{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Step 1: Add and to cancel z.
2x – y + z = 4
x + 3y – z = 11
3x + 2y = 15
1
2
3
1 2
1
2
4
{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Step 1: Subtract and to cancel z.
x + 3y – z = 11
4x + y – z = 14
-3x + 2y = -3
1
2
3
2 3
2
3
5
{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Step 2: Use equations and to find x and y.
3x + 2y = 15
-3x + 2y = -3
4y = 12
y = 3
1
2
3
54
4
5
{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Step 2: Use equations and to find x and y.
3x + 2y = 15
3x + 2(3) = 15
3x + 6 = 15
3x = 9
x = 3
1
2
3
54
Sub y = 3 in 4
4
{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Step 3: Solve for the remaining unknown.
2x – y + z = 4
2(3) – 3 + z = 4
6 – 3 + z = 4
z = 4 + 3 – 6
z = 1
1
2
3
Sub x = 3 and y = 3 in 1
1
{
1) Solving Three-Variable Systems by Elimination
Example 2:
Solve by elimination. 2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
Therefore, the solution is (3, 3, 1).
1
2
3{
Example 3:
Solve by elimination. -x + 2z = -9
-x – 3y – 4z = 2
-3x – 2y + 2z = 17
1) Solving Three-Variable Systems by Elimination
{
Example 3:
Solve by elimination. -x + 2z = -9
-x – 3y – 4z = 2
-3x – 2y + 2z = 17
Therefore, no unique solution.
1) Solving Three-Variable Systems by Elimination
{
Summary of Steps
1) Elimination twice, create equations (4) and (5)
2) Solve for unknown a
3) Substitute a into equation (4) or (5) to find b
4) Substitute a and b into (1), (2) or (3) to find c
Homework
p.157 #1-3, 25, 26