Answer
$I_1=\frac{10}{71},$ $I_3=\frac{55}{71}$ and $I_2=\frac{65}{71}$
Work Step by Step
$\begin{cases} I_2=I_1 + I_3\\
5-3I_1 -5I_2=0\\
10-5I_2-7I_3=0
\end{cases}$
Replacing $I_2$ in the second and third equation by $I_1+I_3$.
$\begin{cases}8I_1+5I_3=5\\
5I_1+12I_3=10
\end{cases}$
Multiplying the first equation by $-\frac{12}{5}$ and adding it to the first equation.
$\begin{cases}-\frac{12}{5}(8I_1+5I_3=5)\\
5I_1+12I_3=10
\end{cases}$
$-\frac{71}{5}I_1=-2,$
$I_1=\frac{10}{71},$ $I_3=\frac{55}{71}$ and $I_2=\frac{65}{71}$
