Solving a simple system of equations

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You should be using Groebner bases (and the Singular interface) instead of general purpose symbolics (via Maxima) for systems of polynomial equations. The standard way of doing what you want is to compute a lexicographic Groebner basis along the lines of what I commented here.

I think your original question was edited, because when I answered before there were the lines

f2 = (c(z+w)(ha-x-z))-(cz*(x+y)) - (e*z)
f3 = (c(x+y)(hb-y-w))-(cy*(z+w)) - (e*y)

I added multiplication signs between cz and cy and pasted the following code into the cell

x,y,z,w,ha,hb,e,c = var('x y z w ha hb e c')
f1 = (c(x+y)(ha-x))-(e*x)
f2 = (c(z+w)(ha-x-z))-(c*z(x+y)) - (e*z)
f3 = (c(x+y)(hb-y-w))-(c*y(z+w)) - (e*y)
f4 = (c(z+w)(hb-w))-(e*w)
solve([f1==0,f2==0,f3==0,f4==0],x,y,z,w)

I then get the answer

[[x == (c*e*hb - e^2*hb + (2*c^2 - e^3 + (c^2 + c)*e - e^2 + 3*c +
1)*ha)/(4*c^2 - e^4 + (c^2 - 1)*e^2 - 2*e^3 + 2*(2*c^2 + c)*e + 4*c +
1), y == -(e^3*hb - (c*hb - hb)*e^2 - c*e*hb - (c*e^2 - (c^2 - 2*c -
1)*e - 2*c^2 - c)*ha)/(4*c^2 - e^4 + (c^2 - 1)*e^2 - 2*e^3 + 2*(2*c^2 +
c)*e + 4*c + 1), z == (c*e^2*hb - 2*c^2*hb - (c^2*hb - 2*c*hb - hb)*e +
((c - 1)*e^2 - e^3 + c*e)*ha - c*hb)/(4*c^2 - e^4 + (c^2 - 1)*e^2 -
2*e^3 + 2*(2*c^2 + c)*e + 4*c + 1), w == -(e^3*hb - 2*c^2*hb + e^2*hb -
(c*e - e^2)*ha - (c^2*hb + c*hb)*e - 3*c*hb - hb)/(4*c^2 - e^4 + (c^2 -
1)*e^2 - 2*e^3 + 2*(2*c^2 + c)*e + 4*c + 1)]]

I just see that * sign is removed from normal text postings on this site, so your original system probably contained all *. In the current form with all * I get no solution either. - sorry

PS: Question to the site admins: Is it a bug or a feature of this site that * is removed?

I replaced cz with c * z and cy with c * y and got a (somewhat lengthy) solution immediatly. Does this solve it?