x =.; p1 =.; p2 =.; y =.; m =.;
m1 =.; m2 =.;
R =.;
R1 =.; R2 =.;
[Rho]1 =.; [Rho]2 =.;
[Sigma] =.;

(*The equations*)
Print["equation1"];
eq1 = x p1 m1 + y p2 m2 - m;
Print["0 = ", eq1];
Print["equation2"];
eq2 = x R1 m1 + y R2 m2;
Print["0 = ", eq2];


a = FullSimplify[Solve[{eq1 == 0, eq2 == 0}, {m1, m2}]];
Print["solution:  ", FullSimplify[a]];

{-(p1/p2) ((m)/x y)/(p1 R2 -  p2 R1) == [Sigma] ,
- R2 y p1/p2 == [Rho]1  ,
 R1 x p1/p2 == [Rho]2