I have found a node equation:
$$ I_3 = I_1 + I_2 $$
I have found two loop equations:
$$ 50 = I_3 + R_1 \times I_1 $$
$$ 50 = I_3 + R_2 \times I_2 $$
I'm not sure what the third loop equation is.
Your two loop equations are incorrect. I3 should not appear in either equation. For one, the units don't match: I3 is a current while R1*I1 and R2*R2 are voltages.Since your have 3 unknowns and 3 equations, a third loop equation is unnecessary. You could create a third loop equation using the loop around R1 and R2 but it does not yield any additional information about the unknown quantities: I1, I2 and I3.