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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.

 Sep 9, 2018

Best Answer 

 #1
avatar+27908 
+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

 Sep 9, 2018
 #1
avatar+27908 
+1
Best Answer

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
avatar+1177 
+1

Correct!

Lightning  Sep 9, 2018

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