Solving three equations

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What this lesson covers

  • Eliminate one variable from two pairs.
  • Solve the resulting two equations.
  • Substitute back and check.
Worked example

x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2

  • Number them (1), (2) and (3).
  1. (1) + (3): 2x + 3y = 8
  2. (2) − (1): x − 2y = −3
  3. so x = 2y − 3
  4. 2(2y − 3) + 3y = 8, so y = 2
  5. x = 1
  6. z = 6 − 1 − 2 = 3

Common mistakes

  • Eliminating different variables from each pair.
    Remove the same variable both times.
  • Skipping the check.
    Test all three equations.

Your turn

  • 1. Solve x + y + z = 9, x − y + z = 3, x + y − z = 1.
Show answer

1. x = 2, y = 3, z = 4

Summary

What to take away

  • Three equations down to two, then to one.
  • Check in all three.

More in AS 91587

Solving three equationsNotesGaussian eliminationNotesUnique, infinite and no solutionsNotesModelling with three unknownsNotes

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