Add up the first four equations and simplify:
p(x+y+z) + p + q(x+y+z) + q + r(x+y+z) + r = 1 + x+y+z
Simplify further:
(p+q+r)(x+y+z) + p+q+r = 1 + x+y+z
Knowing that p+q+r = -3 we can write the above as
-3(x+y+z) - 3 = 1 + x+y+z
Rearranging, we have
-4 = 4(x+y+z) or x+y+z = -1