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For the system of equations 2x + 3y = -1, 3x + y = 2, and lambda*x + 2y = mu to be consistent, which of the following must hold?

1. lambda - mu = 2
2. lambda + mu = -1
3. lambda + mu = 3
4. lambda - mu + 8 = 0

Correct answer: lambda - mu = 2

Solution

From the first two equations: 2x + 3y = -1... (1) and 3x + y = 2... (2). Multiply (2) by 3: 9x + 3y = 6. Subtract (1): 7x = 7, x = 1. From (2): 3 + y = 2, y = -1. For consistency, (x=1, y=-1) must satisfy the third equation: lambda*1 + 2*(-1) = mu => lambda - 2 = mu => lambda - mu = 2... Wait: lambda - mu = 2, which matches option (a)? Let me recheck: lambda*1 + 2*(-1) = mu => lambda - 2 = mu => lambda - mu = 2. That's option A. But let me verify option D: lambda - mu + 8 = 0 => lambda - mu = -8. This doesn't match. So the answer should be lambda - mu = 2 (option A).

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