+1245
Given \(\begin{align*} px+qy+rz&=1,\\ p+qx+ry&=z,\\ pz+q+rx&=y,\\ py+qz+r&=x,\\ p+q+r&=-3, \end{align*} \) find x+y+z.
+33661 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
+33661 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
