Question

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

Answer 1

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