You can want to do anything you want. But unless you have a magic wand, that often fails.
Here, you have ONE equation (3) that relates x, y, and z. ONLY one.
Yes, you have some other equations, relating other variables. Those other equations are irrelevant in this case.
You can find z in terms of x and y. That is trivially easy. (Or x in terms of y and z, etc.) Pencil and paper will suffice. But nothing you will do will allow you to extract z only as a function of the other parameters, in a form that does not include x and y. They are inextricably linked, because only one equation relates x, y, and z.
Would you like a simple example to show this does indeed fail? The following system of 2 LINEAR equations in the 2 unknowns x and y, is comparable to what you wnt to do. See what happens when I pose that to solve.
[xsol,ysol] = solve(a + b == 1,a-b+x+y == 2,x,y)
As you can see, it cannot extract x and y.
Now, suppose I change the question? Can I solve for a and b, separately? Here, as functions of x and y?
[asol,bsol] = solve(a + b == 1,a-b+x+y == 2,a,b)
Do you see why the latter case is different? I now have two equations relating two unknowns. Both equations provide information about the variables in question. Now I can extract a and b separately, unlinking them.
In the first case, we had only the one equation, and there was no solution, no magic wand can exist there. Do you see that even though they are simple linear equations, no algebra you can do will extract x and y separately. Feel free to try. At best, we can solve for x as a function of y, or y as a function of x.
And that is exactly what you are asking to do. But you cannot.