Solve the linear system using the Combination Method

1
Solve the linear system using the Combination
Method

• 4x 4y 8
• 6x 3y 12

2
3.6 Solving Systems of LinearEquations in Three
Variables

• Goal
• Solve systems of linear equations in three
  variables.

3
Solution of a System in 3-Variables

• An ordered triple (x, y, z) that is a SOLUTION of
  ALL THREE equations is a solution to a system of
  equations in three variables.

4
Systems of Equations
3 planes and 3 variables
2 lines and 2 variables
Both
of equations of variables (unknowns)
1 Solution point Infinite Solutions line
5
Solutions to Systems
No Solution All three planes never intersect.
6
Solutions to Systems
One Solution Planes Intersect at one point
. Infinitely Many Solutions Planes Intersect
along a line.
7
Linear Combination Method for a 3-Variable System

• Use the linear combination method to rewrite the
  linear system in 3 variables as a linear system
  in 2 variables.
• Solve the new linear system in 2 variables for
  both of its variables.
• Substitute the values found in step 2 into one of
  the original equations to solve for the remaining
  variable.

8
Solve the system.

• 1. x 2y 3z -2
• 2x 2y z 7
• x y 2z -4

9
Solve the system.

• 2. 3x 2y 4z 11
• 2x y 3z 4
• 5x - 3y 5z -1

10
Solve the system.

• 3. x y z 2
• 3x 3y 3z 14
• x - 2y z 4

11
Solve the system.

• 4. x y z 2
• x y - z 2
• 2x 2y z 4

12
Solve the system.
5. x y z 24 5x 3y z 56 x y z

13
Homework

• Pg. 182 24-29