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Given


px+qy+rz=1,
p+qx+ry&=z,

pz+q+rx&=y,
py+qz+r&=x
p+q+r=-3,

 

Find x+y+z.
 

 Nov 6, 2020
 #1
avatar+25595 
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Given
\(\begin{array}{|rcll|} \hline px+qy+rz&=&1 \\ p+qx+ry&=&z \\ pz+q+rx&=&y \\ py+qz+r&=&x \\ p+q+r &=&-3 \\ \hline \end{array}\)

 

Find x+y+z.

 

\(\small{ \begin{array}{|lrcll|} \hline &px+qy+rz&=&1 \\ &p+qx+ry&=&z \\ &pz+q+rx&=&y \\ &py+qz+r&=&x \\ \hline \text{sum} & (px+py+pz) +(qx+qy+qz) \\ & +(rx+ry+rz) + (p+q+r) \\ & &=& 1+x+y+z \\\ & p(x+y+z) +q(x+y+z) \\ & +r(x+y+z) + (p+q+r) \\ & &=& 1+x+y+z \\ & (x+y+z)(p+q+r)+ (p+q+r) &=& 1+x+y+z \\ & (p+q+r)+(x+y+z)(p+q+r) &=& 1+x+y+z \\ & (p+q+r)(1+x+y+z) &=& 1+x+y+z \quad | \quad p+q+r=-3 \\ & -3(1+x+y+z) &=& (1+x+y+z) \quad | \quad -(1+x+y+z) \\ & -4(1+x+y+z) &=& 0 \quad | \quad : (-4) \\ & 1+x+y+z &=& 0 \\ & \mathbf{x+y+z} &=& \mathbf{-1} \\ \hline \end{array} }\)

 

laugh

 Nov 7, 2020

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