3.6 Solving Systems of Linear Equations in 3 Variables

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Learning Target

• I can solve systems of linear equations in 3 variables.

A system of lin. eqns. in 3 variablesA system of lin. eqns. in 3 variablesLooks something like:Looks something like:

x+3y-z=-11x+3y-z=-112x+y+z=12x+y+z=1

5x-2y+3z=215x-2y+3z=21

A solution is an A solution is an ordered tripleordered triple (x,y,z) that makes all 3 (x,y,z) that makes all 3 equations true.equations true.

Steps for solving in 3 variablesSteps for solving in 3 variables

1.1. Using the 1Using the 1stst 2 equations, cancel one of the 2 equations, cancel one of the variables.variables.

2.2. Using the last 2 equations, cancel the same variable Using the last 2 equations, cancel the same variable from step 1.from step 1.

3.3. Use the results of steps 1 & 2 to solve for the 2 Use the results of steps 1 & 2 to solve for the 2 remaining variables.remaining variables.

4.4. Plug the results from step 3 into one of the original Plug the results from step 3 into one of the original 3 equations and solve for the 33 equations and solve for the 3rdrd remaining variable. remaining variable.

5.5. Write the solution as an ordered triple (x,y,z).Write the solution as an ordered triple (x,y,z).

Solve the system.Solve the system.1.1. x+3y-z=-11x+3y-z=-11

2x+y+z=12x+y+z=1z’s are easy to cancel!z’s are easy to cancel!

3x+4y=-103x+4y=-102. 2x+y+z=12. 2x+y+z=1

5x-2y+3z=215x-2y+3z=21Must cancel z’s again!Must cancel z’s again!

-6x-3y-3z=-3-6x-3y-3z=-35x-2y+3z=215x-2y+3z=21 -x-5y=18-x-5y=18

2(2)+(-4)+z=12(2)+(-4)+z=1 4-4+z=14-4+z=1

3. 3x+4y=-103. 3x+4y=-10 -x-5y=18-x-5y=18

Solve for x & y.Solve for x & y.3x+4y=-103x+4y=-10-3x-15y+54-3x-15y+54

-11y=44-11y=44 y=- 4y=- 4

3x+4(-4)=-103x+4(-4)=-10 x=2x=2

(2, - 4, 1)(2, - 4, 1)

x+3y-z=-11x+3y-z=-112x+y+z=12x+y+z=15x-2y+3z=215x-2y+3z=21

z=1z=1

Solve the system.Solve the system.

1.1. -x+2y+z=3-x+2y+z=32x+2y+z=52x+2y+z=5

z’s are easy to cancel!z’s are easy to cancel!-x+2y+z=3-x+2y+z=3-2x-2y-z=-5-2x-2y-z=-5-3x=-2-3x=-2x=2/3x=2/3

2.2. 2x+2y+z=52x+2y+z=54x+4y+2z=64x+4y+2z=6

Cancel z’s again.Cancel z’s again.-4x-4y-2z=-10-4x-4y-2z=-104x+4y+2z=64x+4y+2z=6 0=- 40=- 4

Doesn’t make sense!Doesn’t make sense!No solutionNo solution

-x+2y+z=3-x+2y+z=32x+2y+z=52x+2y+z=54x+4y+2z=64x+4y+2z=6

Solve the system.Solve the system.1.1. -2x+4y+z=1-2x+4y+z=1

3x-3y-z=23x-3y-z=2z’s are easy to cancel!z’s are easy to cancel!

x+y=3x+y=32.2. 3x-3y-z=23x-3y-z=2

5x-y-z=85x-y-z=8Cancel z’s again.Cancel z’s again.

-3x+3y+z=-2-3x+3y+z=-25x-y-z=85x-y-z=82x+2y=62x+2y=6

3. x+y=33. x+y=32x+2y=62x+2y=6

Cancel the x’s.Cancel the x’s.-2x-2y=-6-2x-2y=-62x+2y=62x+2y=6 0=00=0

This is true.This is true.¸ ¸ many solutionsmany solutions

-2x+4y+z=1-2x+4y+z=13x-3y-z=23x-3y-z=25x-y-z=85x-y-z=8

Assignment