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The system 2x + y + z = 1, x - 2y + z = 2, 3x - y + 2z = 3 has infinitely many solutions. Which statement about these solutions is correct?

1. The solution set is { (-3t - 1, t, 5t + 3): t in R }
2. The solution set is { (3t, -1 - t, -5t + 3): t in R }
3. Every solution satisfies x - 3y + 1 = 0
4. No solution satisfies 5x + 3z = 1

Correct answer: The solution set is { (-3t - 1, t, 5t + 3): t in R }

Solution

eq3 = eq1 + eq2, so two independent equations remain; solving and setting y = t gives the family (-3t-1, t, 5t+3).

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