Answer
$\left( 2,3,5 \right)$
Work Step by Step
According to Cramer’s rule
$x=\frac{{{D}_{x}}}{D}$, $y=\frac{{{D}_{y}}}{D}$, $z=\frac{{{D}_{z}}}{D}$.
$D$ is the determinant and all three denominators consist of x, y and z co-efficients.
$D=$ $\left| \begin{matrix}
1 & -3 & 1 \\
1 & 2 & 0 \\
2 & -1 & 0 \\
\end{matrix} \right|$
${{D}_{x}}$ is the determinant in the numerator for x obtained by replacing the x- co-efficient in D with the constants on the right sides of the equations.
${{D}_{x}}=$ $\left| \begin{matrix}
-2 & -3 & 1 \\
8 & 2 & 0 \\
1 & -1 & 0 \\
\end{matrix} \right|$
${{D}_{y}}$ is the determinant in the numerator for y obtained by replacing the y- co-efficient in D with the constants on the right sides of the equations.
${{D}_{y}}=$ $\left| \begin{matrix}
1 & -2 & 1 \\
1 & 8 & 0 \\
2 & 1 & 0 \\
\end{matrix} \right|$
${{D}_{z}}$ is the determinant in the numerator for z obtained by replacing the z- co-efficient in D with the constants on the right sides of the equations.
${{D}_{z}}=$ $\left| \begin{matrix}
1 & -3 & -2 \\
1 & 2 & 8 \\
2 & -1 & 1 \\
\end{matrix} \right|$.
Calculate the four determinants.
$\begin{align}
& D=\left| \begin{matrix}
1 & -3 & 1 \\
1 & 2 & 0 \\
2 & -1 & 0 \\
\end{matrix} \right| \\
& =1\left( -1-4 \right) \\
& =-5
\end{align}$
$\begin{align}
& {{D}_{x}}=\left| \begin{matrix}
-2 & -3 & 1 \\
8 & 2 & 0 \\
1 & -1 & 0 \\
\end{matrix} \right| \\
& =1\left( -8-2 \right) \\
& =-10
\end{align}$
$\begin{align}
& {{D}_{y}}=\left| \begin{matrix}
1 & -2 & 1 \\
1 & 8 & 0 \\
2 & 1 & 0 \\
\end{matrix} \right| \\
& =1\left( 1-16 \right) \\
& =-15
\end{align}$
$\begin{align}
& {{D}_{z}}=\left| \begin{matrix}
1 & -3 & -2 \\
1 & 2 & 8 \\
2 & -1 & 1 \\
\end{matrix} \right| \\
& =1\left( 2+8 \right)+3\left( 1-16 \right)-2\left( -1-4 \right) \\
& =-25
\end{align}$
Substitute the given values
$\begin{align}
& x=\frac{{{D}_{x}}}{D} \\
& =\frac{-10}{-5} \\
& =2
\end{align}$
$\begin{align}
& y=\frac{{{D}_{y}}}{D} \\
& =\frac{-15}{-5} \\
& =3
\end{align}$
$\begin{align}
& z=\frac{{{D}_{z}}}{D} \\
& =\frac{-25}{-5} \\
& =5
\end{align}$
Therefore, $\left( x,y,z \right)=\left( 2,3,5 \right)$
