Three variable systems of equations can be solved using elimination or substitution similarly to two variable systems. For elimination, equations are paired to eliminate a variable leaving two equations with two unknowns that can then be solved. For substitution, one equation is solved for one variable in terms of others and substituted into remaining equations to create a system that can be solved. An example application involving budgets shirts is worked through to demonstrate solving a three variable system.

1. 3-5 Systems with
Three Variables
Algebra II Unit3 Linear Systems
© Tentinger

2. Essential Understanding and
Objectives
• Essential Understanding: To solve systems of three equations in
three variables, you can use some of the same algebraic methods
you used to solve systems of two equations in two variable.
• Objectives:
• Students will be able to:
• Solve systems of three variables using elimination
• Solve systems of three variable using substitution

3. Iowa Core Curriculum
• Algebra
• Extends A.REI.6 Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.

4. Three Variable Equations
• Two variable equations represent lines
• Three variable equations represent planes
• Like two variable equations, you can have no solution, one
solution, or infinitely many solutions
• Graphs of solutions
• http://www.mathwarehouse.com/algebra/planes/systems/three-
variable-equations.php
• No solution: no point lies in all three planes
• One Solution: the planes intersect at one common point
• Infinitely Many Solutions: The planes intersect at a line

5. Solving a system using
Elimination
• Step 1: Pair the equations to eliminate one variable, z. Then
you will have two equations with two unknowns.
• AddSubtract

6. Solving a system using
Elimination
• 2: Write the new equations as a system. Solve for x and y
• Add and solve for y.
• Substitute your answer and solve for x

7. Solving a system using
Elimination
• Step 3: Solve for remaining variable, z. Substitute in answers
for x and y into the original equations
• Step 4: Write the solution as an ordered triple: (3, 3, 1)

8. Solve using Elimination

9. Solving Equivalent Systems

10. Solving a System using
Substitution:
• Step 1: chose the equation whose variable is easy to isolate.
• X+5y=9 x = -5y+9
• Step 2: Substitute the expression into the other two remaining
equations and simplify
• 2(-5y+9) + 3y – 2z = -1 4z – 5(-5y+9) = 4
• -7y -2z = -19 25y +4z = 49

11. Solving a System using
Substitution:
• Step 3: Write the two new equations as a system and solve for
the remaining variables
• use elimination to solve for y
then substitute to solve for z
• y = 1, z = 6
• Step 4: Use the original equation to solve for x
• Solution (4, 1, 6)

12. Solve by substitution

13. Application
• You manage a clothing store and budget $5400 to restock 200 shirts.
You can buy T-shirts for $12 each, polo shirts for $24 each, and
rugby shirts for #36 dollars each. If you want to have the same
number of T-shirts as polo shirts, how many of each shirt should
you buy?
• Relate:
• T-shirts + polo shirts + rugby shirts = 200
• T-shirts = polo shirts
• 12 * Tshirts + 24*polo shirts + 36*rugby shirts = 5400
• Define:
• X = tshirts
• Y = polo
• Z = rugby

14. Application
• You manage a clothing store and budget $5400 to restock 200 shirts.
You can buy T-shirts for $12 each, polo shirts for $24 each, and
rugby shirts for #36 dollars each. If you want to have the same
number of T-shirts as polo shirts, how many of each shirt should
you buy?
• Write:
• Solve:
• Substitute x in for equations 1 and 3 then simplify
• Write the new equations as a system then solve for y and z
• Substitute y and z back into one of the original equations to get x
• Solution: (50, 50, 100)

15. Homework
• Pg. 171-172
• #14-16, 24-26, 32, 34-37