Q:

Solve three equations for three unknowns.

So I have the following three equations which I do not know how to solve:
-D * x - E * y = A + (R * D)
E * F * x - D * F * y - G * z = B - (R * E * F)
E * G * x - D * G * y + F * z = C - (R * E * G)
What is the solution for this? How do I get the values of x, y and z. The rest are constants.

A:

You have three linear equations in three unknowns.  There are a number of well known methods for doing that.  Here, the first thing I notice is that there is no "z" in the first equation so it might be simplest to eliminate z from the other two.  The second equation has "-Gz" and the third equation has "Fz". 
Multiplying the second equation by F gives:
E * F^2 * x - D * F^2 * y - F*G * z = B*F - (R * E * F^2)."|

Multiplying the third equation by G gives:
E * G^2 * x - D * G^2* y + F *G* z = C*G - (R * E * G^2).

Now, add the two equations to eliminate z:
E*(F^2+ G^2)x- D*(F^2+ G^2)y=  B*F+ C*G- R*E*(F^2- G^2).  

That, together with the first equation, -D * x - E * y = A + (R * D), gives two equations in two unknowns, x and y, so our next objective would be to eliminate one of those.
Seeing the "E*(F^2+ G^2)" multiplying x in the equation above and "-D" in this one, I would multiply this equation by E*(F^2+ G^2) to get:
-D *E*(F^2+ G^2)*x - E^2*(F^2+ G^2)*y = A*E*(F^2+ G^2) + R * D* E*(F^2+ G^2)

and multiply the previous one by D to get:
D*E*(F^2+ G^2)x- D^2*(F^2+ G^2)y=  B*D*F+ C*D*G- R*D*E*(F^2- G^2).

Now adding those eliminates x leaving a single equation, 
[-E^2*(F^2+ G^2)- D^2(F^2+ G^2)]y= -(E^2+ D^2)*(F^2+ G^2)y= A*E*(F^2+ G^2) + R * D* E*(F^2+ G^2)+ B*D*F+ C*D*G- R*D*E*(F^2- G^2) 

that is easily solved for y.  Putting that value for y back into either of the equations involving only x and y gives an equation that can be solved for x and putting those value for x and y back into either of the equations that involved x, y, and z gives an equation to solve for z.

