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Given

 

px + qy + rz = 1

p + qx + ry = z

pz + q + rx = y

py + qz + r = x

x + y + z = -3

 

find p + q + r

 Apr 18, 2021
 #1
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If you add the first 4 equations:

\(p+q+r+pz+py+pz+qx+qy+qz+rx+ry+rz=x+y+z+1\)

substitute x+y+z for -3:

\(p+q+r+pz+py+pz+qx+qy+qz+rx+ry+rz=-2\\(p+q+r)+p(x+y+z)+q(x+y+z)+r(x+y+z)=-2\\ (p+q+r)+(p+q+r)(x+y+z)=-2\\(p+q+r)(1+(x+y+z))=-2\)

substitute x+y+z for -3 again:

\((p+q+r)(-2)=-2\\\boxed{p+q+r=1}\)

 Apr 18, 2021

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