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 Homework Statement

For the circuit in Figure 1, find the current and voltage at resistor R1 considering that the values of the voltage sources
they are V1 [V] and V2 [V], and the resistance values are R1 [Ω], R2 [Ω] and R3 [Ω] respectively.
 Homework Equations

Kirchhoff's Laws:
##\sum_i I_i=0##
##\sum_i V_i=0##
I tried to solve it by loop currents. So on the left mesh the loop current ##I_1## goes clockwise and on the right mesh the loop current ##I_2## goes counterclockwise.
I ended up with the following equations:
1) ##V_1R_1(I_1+I_2)R_2I_1=0##;
2) ##V_2R_3I_2R_1(I_1+I_2)=0##.
To find the current on ##R_1##, I tried to find ##I_1## and ##I_2##. I first solved 1) for ##I_1##: ##I_1=\frac{V_1R_1I_2}{R_1+R_2}##. Then I solved 2) for ##I_2## replacing the expression that I found for ##I_1## :
##V_2R_3I_2R_1(I_1+I_2)=V_2I_2(R_3R_1)R_1I_1##
##=V_2I_2(R_3R_1)R_1\left(\frac{V_1R_1I_2}{R_1+R_2}\right)=V_2I_2(R_3R_1)R_1\left(\frac{V_1R_1I_2}{R_1+R_2}\right)##
##=V_2(R_1+R_2)I_2(R_3R_1)(R_1+R_2)R_1(V_1R_1I_2)##
##=V_2(R_1+R_2)I_2[(R_3R_1)(R_1+R_2)R_1^2]R_1V_1##
##I_2=\frac{V_2(R_1+R_2)R_1V_1}{(R_1+R_2)(R_3R_1)R_1^2}##
Finally, I replaced ##I_2## on ##I_1##: ##I_1=\frac{V_1R_1\left(\frac{V_2(R_1+R_2)R_1V_1}{(R_1+R_2)(R_3R_1)R_1^2}\right)}{R_1+R_2}##. Am I correct?##=V_2I_2(R_3R_1)R_1\left(\frac{V_1R_1I_2}{R_1+R_2}\right)=V_2I_2(R_3R_1)R_1\left(\frac{V_1R_1I_2}{R_1+R_2}\right)##
##=V_2(R_1+R_2)I_2(R_3R_1)(R_1+R_2)R_1(V_1R_1I_2)##
##=V_2(R_1+R_2)I_2[(R_3R_1)(R_1+R_2)R_1^2]R_1V_1##
##I_2=\frac{V_2(R_1+R_2)R_1V_1}{(R_1+R_2)(R_3R_1)R_1^2}##
I'm not sure how to find the voltage on ##R_1##, but I think that I need to find ##R_1(I_1+I_2)##, right?