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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?