If x, y, w, z are real numbers satisfying
\(w+x+y = -2 \\ w+x+z = 4 \\ w+y+z = 19 \\ x+y+z = 12\)
What is \(wx + yz\)?
\(\begin{array}{|lrcll|} \hline (1)& w+x+y &=& -2 \\ (2)& w+x+z &=& 4 \\ (3)& w+y+z &=& 19 \\ (4)& x+y+z &=& 12 \\ \hline (1)+(2)+(3)+(4): & 3w+3x+3y+3z &=& -2+4+19+12 \\ & 3(w+x+y+z) &=& 33 \quad | \quad : 3 \\ & \mathbf{w+x+y+z} &=& \mathbf{11} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{w+x+y+z} &=& \mathbf{11} \quad | \quad x+y+z=12 \\ w+12 &=& 11 \\ w &=& 11-12 \\ \mathbf{w} &=& \mathbf{-1} \\ \hline \end{array}\\ \begin{array}{|rcll|} \hline \mathbf{w+x+y+z} &=& \mathbf{11} \quad | \quad w+y+z=19 \\ x+19 &=& 11 \\ x &=& 11-19 \\ \mathbf{x} &=& \mathbf{-8} \\ \hline \end{array}\\ \begin{array}{|rcll|} \hline \mathbf{w+x+y+z} &=& \mathbf{11} \quad | \quad w+x+z=4 \\ y+4 &=& 11 \\ y &=& 11-4 \\ \mathbf{y} &=& \mathbf{7} \\ \hline \end{array}\\ \begin{array}{|rcll|} \hline \mathbf{w+x+y+z} &=& \mathbf{11} \quad | \quad w+x+y=-2 \\ z-2 &=& 11 \\ z &=& 11+2 \\ \mathbf{z} &=& \mathbf{13} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline wx + yz &=& (-1)(-8)+7*13 \\ &=& 8+91 \\ \mathbf{wx + yz} &=& \mathbf{99} \\ \hline \end{array}\)
