Answer
$x_{1} = 5$
$x_{2} = 3$
$x_{3} = -1$
Work Step by Step
The procedure is shown with the matrix notation for simpler understanding.
$x_{1} - 3x_{3} = 8 $
$2x_{1} + 2x_{2} + 9x_{3} = 7$
$x_{2} + 5x_{3} = -2$
This can be depicted in the augmented matrix notation as follows :
$\begin{bmatrix}
1 & 0 & -3 & 8 \\
2 & 2 & 9 & 7 \\
0 & 1 & 5 & -2
\end{bmatrix}$
To eliminate the $2x_{1}$ term in the second equation, add $-2$ times row 1 to row 2:
$\begin{bmatrix}
1 & 0 & -3 & 8 \\
0 & 2 & 15 & -9 \\
0 & 1 & 5 & -2
\end{bmatrix}$
To get an $x_{2}$ term in the second row, interchange row 2 and row 3:
$\begin{bmatrix}
1 & 0 & -3 & 8 \\
0 & 1 & 5 & -2\\
0 & 2 & 15 & -9
\end{bmatrix}$
Next we use the $x_{2}$ term in the second equation to eliminate the $2x_{2}$ term from the third equation. Add $-2$ times
row 2 to row 3:
$\begin{bmatrix}
1 & 0 & -3 & 8 \\
0 & 1 & 5 & -2\\
0 & 0 & 5 & -5
\end{bmatrix}$
Now the augmented matrix is in a triangular form. For interpreting it, we go back to the equation notation:
$x_{1} - 3x_{3} = 8$
$x_{2} + 5x_{3} = -2$
$5x_{3} = -5$
From $5x_{3} = -5$, we get $x_{3} = -1$
Using this value in $x_{2} + 5x_{3} = -2$, we get:
$x_{2} + 5(-1) = -2$
or, $x_{2} = 3$
Using the value of $x_{3}$ in the equation $x_{1} - 3x_{3} = 8$,
$x_{1} - 3(-1) = 8$
or, $x_{1} = 5$
Hence,
$x_{1} = 5$ ; $x_{2} = 3$ ; $x_{3} = -1$ .
