Problem $4.(25$pts$.)$ Take a look at this nice circuit shown below, where: $R_1=10,R_2=12,R_3=20,V_1=1\mathrm{.}0V,V_2=7\mathrm{.}0V$

$($a$)(20$ pts$.)$ Write the loop and junction equations needed to determine the currents through the resistors. Indicate clearly, on the drawing, the loop used to determine each loop equation.

$($b$)(5$ pts$.)$ Determine the current through the resistor $R_1.$

Present your solution to the problem in the space below.