Kirchoff's Law Loop Equation Problem
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.