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Consistency of a system and the number of solutions sheet
3-mark answer frame

Examine the consistency of the system x + y + z = 1, x + 2y + 3z = 2, 2x + 3y + 4z = 5.

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

  1. 1 · Write what is given and what is to be found. Given: three linear equations. To find: whether the system is consistent.
  2. 2 · Write the system as AX = B. A = [1, 1, 1; 1, 2, 3; 2, 3, 4], X = [x; y; z], B = [1; 2; 5].
  3. 3 · Find |A|. |A| = 1(8 − 9) − 1(4 − 6) + 1(3 − 4) = −1 + 2 − 1 = 0. So there is no unique solution.
  4. 4 · Find adj A. adj A = [−1, −1, 1; 2, 2, −2; −1, −1, 1].
  5. 5 · Find (adj A)B. (adj A)B = [−1 − 2 + 5; 2 + 4 − 10; −1 − 2 + 5] = [2; −4; 2] ≠ O.
  6. 6 · Check and box the answer. |A| = 0 and (adj A)B ≠ O, so there is no solution. Indeed, adding the first two equations gives 2x + 3y + 4z = 3, not 5. The system is inconsistent.

Scoring tip: State the rule in words: '|A| = 0 and (adj A)B ≠ O, so the system is inconsistent.' The rule earns the final mark.