If the system of equations gathered 2x+7y+ z=3 3x+2y+5z=4 x+ y+32z=-1 gathered has infinit…

Question

If the system of equations

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

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

Answer: 38

Accepted Answer

To determine if the system of equations:

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

has infinitely many solutions, we must use Cramer's rule.

The determinants are calculated as follows:

$\begin{gathered} \Delta =-2\lambda +3\lambda \mu -10\mu -509 \ {\Delta}_{1} =2\lambda +3\lambda \mu -15\mu -739 \ {\Delta}_{2} =-7\lambda -7 \ {\Delta}_{3} =\mu +39 \end{gathered}$

To have infinitely many solutions, the determinants must satisfy:

$\begin{gathered} \Delta={\Delta}_{1}={\Delta}_{2}={\Delta}_{3}=0 \end{gathered}$

Solving these equations, we find:

$\lambda =-1,\mu =-39$

Thus, the value of $\lambda -\mu $ is:

$\lambda -\mu =38$

Solutions · 1