# Find the real solution set of the system of equations, please { <mtable columnal

Find the real solution set of the system of equations, please

$\left\{\begin{array}{l}\frac{1}{z}-\frac{2}{x}=5\\ \frac{3}{x}-\frac{2}{y}=1\\ \frac{1}{z}-\frac{1}{y}=5\end{array}$
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Kayley Jackson
We have the equations:
$\frac{1}{z}-\frac{2}{x}=5$ (1)
$\frac{3}{x}-\frac{2}{y}=1$ (2)
$\frac{1}{z}-\frac{1}{y}=5$ (3)

Subtract equation (3) from equation (1)
$\frac{1}{z}-\frac{2}{x}-\frac{1}{z}+\frac{1}{y}=5-5$
$-\frac{2}{x}+\frac{1}{y}=0$
$\frac{1}{y}=\frac{2}{x}$ (4)

Now, put this value in equation (2)
$\frac{3}{x}-\frac{4}{x}=1$
$-\frac{1}{x}=1$
$x=-1$

Put $x=-1$ in equation (4)
$\frac{1}{y}=\frac{2}{-1}$
$y=-\frac{1}{2}$

Now, put $x=-1$ in equation (1)
$\frac{1}{z}+2=5$
$\frac{1}{z}=3$
$z=\frac{1}{3}$
$\left\{\begin{array}{l}x=-1\\ y=-\frac{1}{2}\\ z=\frac{1}{3}\end{array}$
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