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3-5 Systems with
Three Variables
Algebra II Unit3 Linear Systems
© Tentinger
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
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.
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
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
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
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)
Solve using Elimination
Solving Equivalent Systems
Solving a System using
Substitution:



• Step 1: choose 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
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)
Solve by substitution
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
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)
Homework
• Pg. 171-172
• #14-16, 24-26, 32, 34, 37
• 9 problems

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Alg II 3-5 Sytems Three Variables

  • 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)
  • 10. Solving a System using Substitution: • Step 1: choose 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)
  • 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 • 9 problems

Editor's Notes

  1. Solution: (4, 2, -3)Solution: (2, -1, 4)Solution: (-1, 2, -4)