If your problem is linear then this
{vector,matrix}=CoefficientArrays[{m1,m2,m3},{x,y,z}];
solution=LinearSolve[matrix,vector]
will extract your coefficients from your equations and solve the linear system without needing to know or use Rationalize or N, and this then displays x,y,z
{2.43756*^22,-2.94013*^20,-1.98547*^25}
and then this will substitute that solution into your 3x+2y expression
ReplaceAll[3x+2y,Thread[Rule[{x,y,z},solution]]]
which displays the result
7.25388*^22
If you manually extract the coefficients from your equations and or insert the solution into your expression then you could do this without needing to know most of these functions.