$7-1.$ Consider the two$-$loop circuit shown in Fig. P$7\mathrm{.}1.$ The currents $I_1$ and $I_2($in amps $)$ satisfy the following system of equations:

$-5000I_1+1000I_2=-8$

$1000I_1=10+3000I_2$

$($a$)$ Find $I_1$ and $I_2$ using the substitution method.

$($b$)$ Write the system of equations in the matrix form $AI=b,$ where $I=\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right].$

$($c$)$ Find $I_1$ and $I_2$ using Cramer's rule.

$($d$)$ Find $I_1$ and $I_2$ using the matrix algebra method. Perform all matrix computations by hand, and show all steps.

Figure P$7\mathrm{.}1$ Two$-$loop circuit for problem P$7-1.$