Unique, infinite and no solutions

← Previous3 of 4Next →

What this lesson covers

  • One point: a unique solution.
  • A line: infinitely many solutions.
  • A false statement like 0 = 1 means none.
Worked example

No solution

  • x + y + z = 1 and x + y + z = 2.
  1. Subtract: 0 = 1
  2. Parallel planes, so no solution
Worked example

Infinitely many

  • x + y + z = 6, x − y + z = 2, 2x + 2z = 8.
  1. (1) + (2) gives equation (3) again
  2. (1) − (2): y = 2
  3. x + z = 4
  4. Let z = t: x = 4 − t, y = 2, z = t

Common mistakes

  • Thinking 0 = 0 is an error.
    It shows dependent equations.
  • Giving one value as the answer.
    Infinite solutions need a parameter.

Your turn

  • 1. Classify x + y + z = 2, 2x + y − z = 1, 3x + 2y = 4.
  • 2. Classify the same system with 3x + 2y = 3.
Show answer

1. None: adding the first two gives 3x + 2y = 3 2. Infinitely many

Summary

What to take away

  • A false line means no solution.
  • A row of zeros means a family of solutions.

More in AS 91587

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

Is your child on track?

Take the free placement test