Solve a two-battery circuit

The left branch has a $10\ \text{V}$ battery and a $2\ \Omega$ resistor; the right branch has a $6\ \text{V}$ battery and a $2\ \Omega$ resistor; a $4\ \Omega$ resistor is the middle branch. Both positive terminals are on the top wire. The program already holds Kirchhoff's equations as a matrix. You will give it the right-hand side, check the energy, and name the battery that is being charged.

The idea. Unknowns: $I_1$ up through the left battery, $I_2$ up through the right battery, $I_3$ down the middle. Junction: $I_1 + I_2 - I_3 = 0$. Loops: $R_1 I_1 + R_3 I_3 = E_1$ and $R_2 I_2 + R_3 I_3 = E_2$.

Your tasks.

1. Fill in b, the right-hand side of the three equations, in the same order as the rows of A.
2. Find supplied, the power the two batteries give together, $E_1 I_1 + E_2 I_2$.
3. Set charged to "E1" or "E2": the battery whose current comes out negative.