Gaussian elimination

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What this lesson covers

  • Write the system as an augmented matrix.
  • Row operations keep the same solutions.
  • Back substitute from the bottom row.
Key idea

Row operations

  • Swap rows.
  • Multiply a row by a number.
  • Add a multiple of one row to another.
Worked example

The same system

  • Write only the coefficients.
  1. [1 1 16], [2 −1 1 | 3], [1 2 −1 | 2]
  2. R2 − 2R1: [0 −3 −1−9]
  3. R3 − R1: [0 1 −2−4]
  4. R2 + 3R3: [0 0 −7−21]
  5. z = 3
  6. y − 2(3) = −4, so y = 2
  7. x = 1

Common mistakes

  • Changing only one side of a row.
    Apply the operation across the whole row.
  • Overwriting the row you need next.
    Keep the working tidy, row by row.

Your turn

  • 1. What is the augmented matrix of x + y = 3, x − y = 1?
Show answer

1. [1 1 | 3], [1 −1 | 1]

Summary

What to take away

  • Same maths, fewer letters.
  • Aim for zeros under the diagonal.

More in AS 91587

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

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