5.3 Systems of Linear Equations in Three Variables

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5.3 Systems of Linear Equations in Three
Variables
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• A of a system of linear equations in three
  variables is an
• of real numbers that satisfies all
  equations of the system.
• In other words, the solution is the point of
• of three planes.

solution
ordered
triple
intersection
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Solving Linear Combination Method

• 1. Use the linear combination method to rewrite
  the linear system in three variables as a linear
  system in two variables.
• Combine two of the equations to eliminate one
  variable, then combine another two equations to
  eliminate the same variable.
• 2. Solve the new linear system for both of its
  variables.
• 3. Substitute the values found in step 2 into one
  of the original equations and solve for the
  remaining variable.

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Solve the system of linear equations.

• 1.

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Solve the system of linear equations.

• 2.

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• A system of linear equations in three variables
  can also have solutions. This happens when
  one of the following
• 1. three different planes intersect in a single
  line
• 2. two coinciding planes intersect in a third
  plane
• 3. three coinciding planes.

infinite
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• A system of linear equations in three variables
  can also have solution. This happens when one
  of the following
• 1. three parallel planes
• 2. two coinciding planes parallel to a third
  plane
• 3. three planes intersecting in three parallel
  lines
• 4. two parallel lines intersecting a third plane
  in two parallel lines

no
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Solve the system of linear equations.

• 3.

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Solve the system of linear equations.

• 4.