Question

If the system of equations \[2x + 7y + \lambda z = 3,\]\[3x + 2y + 5z = 4,\]\[x + \mu y + 32z = -1\]has infinitely many solutions, then $(\lambda - \mu)$ is equal to ________.

Answer 1

Correct Answer: 38

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. We set:
\[D = D_1 = D_2 = D_3 = 0\]
Calculating \( D_3 \):
\[D_3 = \begin{vmatrix}2 & 7 & 3 \\3 & 2 & 4 \\1 & \mu & -1\end{vmatrix}= 0\]

Expanding, we get:
\[2 \begin{vmatrix}2 & 4 \\\mu & -1\end{vmatrix}- 7 \begin{vmatrix}3 & 4 \\1 & -1\end{vmatrix}+ 3 \begin{vmatrix}3 & 2 \\1 & \mu\end{vmatrix}= 0\]
Solving for \( \mu \), we find:
\[\mu = -39\]
Now, calculating \( D \) with \( \lambda \) in place:
\[D = \begin{vmatrix}2 & 7 & \lambda \\3 & 2 & 5 \\1 & -39 & 32\end{vmatrix}= 0\]
Solving this determinant, we get:
\[\lambda = -1\]

Thus, \( \lambda - \mu = -1 - (-39) = 38 \).