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Let a, lambda, mu be real numbers and consider the system a*x + 2y = lambda, 3x - 2y = mu. Which of the following statements is correct?

1. If a = -3, then the system has infinitely many solutions for all values of lambda and mu
2. If a is not equal to -3, then the system has a unique solution for all values of lambda and mu
3. If lambda + mu = 0, then the system has infinitely many solutions for a = -3
4. If a = -3 and lambda + mu is nonzero, then the system has no solution

Correct answer: If a is not equal to -3, then the system has a unique solution for all values of lambda and mu

Solution

The coefficient determinant equals -2(a + 3), which is nonzero whenever a is not -3, guaranteeing a unique solution for all lambda, mu; when a = -3 the system is consistent (infinitely many solutions) only if lambda + mu = 0, otherwise inconsistent.

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