For which of the following ordered pairs (μ, δ), the system of linear equations <br>x + 2y + 3z = 1<br>3x + 4y + 5z = μ<br>4x + 4y + 4z = δ<br>is inconsistent?
Step-by-Step Solution
Key Concept: A system of linear equations is inconsistent if the determinant of the coefficient matrix is zero and the augmented matrix has a rank different from the coefficient matrix, or more simply, if the equations lead to a contradiction like 0 = k (where k is non-zero).
The system is: <br>x + 2y + 3z = 1 (Eq 1)<br>3x + 4y + 5z = \mu (Eq 2)<br>4x + 4y + 4z = \delta (Eq 3)<br>Subtracting (Eq 1) from (Eq 2): 2x + 2y + 2z = \mu - 1, so x + y + z = (\mu - 1)/2.<br>From (Eq 3), 4(x + y + z) = \delta, so x + y + z = \delta/4.<br>For inconsistency, (\mu - 1)/2 \neq \delta/4, or if the system is dependent, we check for contradictions. Specifically, if we perform row operations: R2 -> R2 - 3R1 and R3 -> R3 - 4R1, we get:<br>x + 2y + 3z = 1<br>-2y - 4z = \mu - 3<br>-4y - 8z = \delta - 4<br>For inconsistency, the ratio of coefficients of y and z must be equal, but the ratio of constants must be different. Here, -4/-2 = -8/-4 = 2. So we need (\delta - 4) \neq 2(\mu - 3).<br>Checking option (4): \mu=4, \delta=3. \delta - 4 = 3 - 4 = -1. 2(\mu - 3) = 2(4 - 3) = 2. Since -1 \neq 2, the system is inconsistent.
Correct Answer: (4)