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Solving systems by row reduction (Gauss-Jordan) sheet
Essay frame

Solve by Gauss-Jordan elimination: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write the augmented matrix. [1 1 1 | 6; 2 −1 1 | 3; 1 2 −1 | 2].
  2. 2 · Clear column 1. R2 → R2 − 2R1, R3 → R3 − R1: [1 1 1 | 6; 0 −3 −1 | −9; 0 1 −2 | −4].
  3. 3 · Make the second leading 1. Swap R2 and R3: [1 1 1 | 6; 0 1 −2 | −4; 0 −3 −1 | −9].
  4. 4 · Clear column 2. R1 → R1 − R2, R3 → R3 + 3R2: [1 0 3 | 10; 0 1 −2 | −4; 0 0 −7 | −21].
  5. 5 · Third leading 1 and clear column 3. R3 → (−1/7)R3, then R1 → R1 − 3R3, R2 → R2 + 2R3: [1 0 0 | 1; 0 1 0 | 2; 0 0 1 | 3].
  6. 6 · Read and check. x = 1, y = 2, z = 3; all three equations check.