3.6 Solving Systems of Linear Equations in 3 Variables

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3.6 Solving Systems of Linear Equations in 3
Variables

• p. 177

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Learning Target

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

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A system of lin. eqns. in 3 variables

• Looks something like
• x3y-z-11
• 2xyz1
• 5x-2y3z21
• A solution is an ordered triple (x,y,z) that
  makes all 3 equations true.

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Steps for solving in 3 variables

1. Using the 1st 2 equations, cancel one of the
   variables.
2. Using the last 2 equations, cancel the same
   variable from step 1.
3. Use the results of steps 1 2 to solve for the 2
   remaining variables.
4. Plug the results from step 3 into one of the
   original 3 equations and solve for the 3rd
   remaining variable.
5. Write the solution as an ordered triple (x,y,z).

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Solve the system.
x3y-z-11 2xyz1 5x-2y3z21

• x3y-z-11
• 2xyz1
• zs are easy to cancel!
• 3x4y-10
• 2. 2xyz1
• 5x-2y3z21
• Must cancel zs again!
• -6x-3y-3z-3
• 5x-2y3z21
• -x-5y18
• 2(2)(-4)z1
• 4-4z1

• 3. 3x4y-10
• -x-5y18
• Solve for x y.
• 3x4y-10
• -3x-15y54
• -11y44
• y- 4
• 3x4(-4)-10
• x2
• (2, - 4, 1)

z1
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Solve the system.
-x2yz3 2x2yz5 4x4y2z6

• 2x2yz5
• 4x4y2z6
• Cancel zs again.
• -4x-4y-2z-10
• 4x4y2z6
• 0- 4
• Doesnt make sense!
• No solution

• -x2yz3
• 2x2yz5
• zs are easy to cancel!
• -x2yz3
• -2x-2y-z-5
• -3x-2
• x2/3

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Solve the system.
-2x4yz1 3x-3y-z2 5x-y-z8

• -2x4yz1
• 3x-3y-z2
• zs are easy to cancel!
• xy3
• 3x-3y-z2
• 5x-y-z8
• Cancel zs again.
• -3x3yz-2
• 5x-y-z8
• 2x2y6

• 3. xy3
• 2x2y6
• Cancel the xs.
• -2x-2y-6
• 2x2y6
• 00
• This is true.
• many solutions

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Assignment
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