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# Help!

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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.

Lightning  Sep 9, 2018

#1
+26971
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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

Alan  Sep 9, 2018
#1
+26971
+1

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

Alan  Sep 9, 2018
#2
+701
+1

Correct!

Lightning  Sep 9, 2018