system

system

$x(y + z) = 39\\ y(x + z) = 60\\ z(x + y) = 49$
What is $\mathbf{x}$?

$\begin{array}{|lrcll|} \hline & x(y + z) &=& 39 \\ (1) & xy+xz &=& 39 \\\\ &y(z + x) &=& 60 \\ (2)&yz + yx &=& 60 \\\\ &z(x + y) &=& 49 \\ (3) & zx+zy &=& 49\\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (1)-(2)+(3): &xy+xz-yz-yx +zx+zy&=& 39 -60+49 \\ &2xz &=& 28 \\ (4) &\mathbf{xz} &=& \mathbf{14} \\\\ (1)+(2)-(3): &xy+xz+yz+yx -zx-zy&=& 39 +60-49 \\ &2xy &=& 50 \\ (5) &\mathbf{xy} &=& \mathbf{25} \\\\ -(1)+(2)+(3): &-xy-xz+yz+yx +zx+zy&=& -39 +60+49 \\ &2yz &=& 70 \\ (6) &\mathbf{yz} &=& \mathbf{35} \\\\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline \frac{(6)}{(4)}: & \dfrac{yz}{xz}&=&\dfrac{35}{14} \\\\ & \dfrac{y}{x}&=&\dfrac{35}{14} \\\\ & y &=&\dfrac{5}{2}*x \\\\ (5) & xy &=& 25\\\\ & x*\dfrac{5}{2}*x&=& 25 \\\\ & x^2 &=& \dfrac{2*25}{5} \\ & x^2 &=& 10 \\ & \mathbf{x} &=& \mathbf{\sqrt{10} }\\ \hline \end{array}$