Solving Systems of Linear Equations in 3 Variables

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Solving Systems of Linear Equations in 3 Variables
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Pictured below is an example of the graph of the
plane 2x 3y z 6. The red triangle is the
portion of the plane when x, y, and z values are
all positive. This plane actually continues off
in the negative direction. All that is pictured
is the part of the plane that is intersected by
the positive axes (the negative axes have dashed
lines).
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Like systems of linear equations, the solution of
a system of planes can be no solution, one
solution or infinite solutions.
No Solution of three variable systems There is no
single point at which all three planes
intersect, therefore this system has no solution.
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One Solution of three variable systems If three
planes intersect, then the three variable system
has 1 point in common, and a single solution
represented by the black point below.
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Infinite Solutions of three variable systems If
the three planes intersect, then the three
variable system has a line of intersection and
therefore an infinite number of solutions.
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A system of linear equations in 3 variables looks
something like

• x 3y z -11
• 2x y z 1
• 5x 2y 3z 21

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

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Is (1, 1, 1) a solution?

• Yes

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Is (2, 1, 6) a solution?

• No

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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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1. Solve the system.
? ? ?

• (2, -4, 1)

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2. Solve the system.
? ? ?

• (1, 1, 3)

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3. Solve the system.
? ? ?

• No solution

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4. Solve the system.
? ? ?

• (-1, 0, 5)

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5. Solve the system.
? ? ?

• (-2, -8, -1)