Modelling with three unknowns

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What this lesson covers

  • Name three unknowns.
  • Each fact gives one equation.
  • A parabola through three points is a system.
Worked example

Tickets

  • 100 tickets earn $1600. Adults pay $20, children $10 and seniors $15, with twice as many adults as children.
  1. a + c + s = 100
  2. 20a + 10c + 15s = 1600
  3. a = 2c
  4. s = 100 − 3c
  5. 40c + 10c + 1500 − 45c = 1600
  6. c = 20
  7. 20 children, 40 adults, 40 seniors
Worked example

Parabola through (1, 2), (2, 3), (3, 6)

  • Use y = ax² + bx + c.
  1. a + b + c = 2
  2. 4a + 2b + c = 3
  3. 9a + 3b + c = 6
  4. 3a + b = 1 and 5a + b = 3
  5. a = 1, b = −2, c = 3
  6. y = x² − 2x + 3

Common mistakes

  • Reusing the letter c for two things.
    Pick different letters for each unknown.
  • Stopping at the letters.
    Answer in context.

Your turn

  • 1. Find the parabola through (0, 1), (1, 2) and (2, 5).
Show answer

1. y = x² + 1

Summary

What to take away

  • One fact, one equation.
  • Check with the original story.

More in AS 91587

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

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