If the system of equations

$$\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to ______ :

<p>To determine if the system of equations:</p> <p>$$\begin{aligned} 2x + 7y + \lambda z = 3 \\ 3x + 2y + 5z = 4 \\ x + \mu y + 32z = -1 \end{aligned}$$</p> <p>has infinitely many solutions, we must use Cramer's rule.</p> <p>The determinants are calculated as follows:</p> <p>$$\begin{aligned} \Delta &= -2\lambda + 3\lambda\mu - 10\mu - 509 \\ \Delta_1 &= 2\lambda + 3\lambda\mu - 15\mu - 739 \\ \Delta_2 &= -7\lambda - 7 \\ \Delta_3 &= \mu + 39 \end{aligned}$$</p> <p>To have infinitely many solutions, the determinants must satisfy:</p> <p>$$\begin{aligned} \Delta = \Delta_1 = \Delta_2 = \Delta_3 = 0 \end{aligned}$$</p> <p>Solving these equations, we find:</p> <p>$\lambda = -1, \mu = -39$</p> <p>Thus, the value of $ \lambda - \mu $ is:</p> <p>$\lambda - \mu = 38$</p>