Intermediate Algebra, Chapter 4, 4.2, Section 4.2, Problem 14

Solve the system of equations $
\begin{equation}
\begin{aligned}

x + 2y + 3z =& 1 \\
-x - y + 3z =& 2 \\
-6x + y + z =& -2

\end{aligned}
\end{equation}
$.


$
\begin{equation}
\begin{aligned}

x + 2y + 3z =& 1
&& \text{Equation 1}
\\
-x - y + 3z =& 2
&& \text{Equation 2}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\phantom{x -} y + 6z =& 3
&& \text{Add}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

6x + 12y + 18z =& 6
&& 6 \times \text{Equation 1}
\\
-6x + y + z =& -2
&& \text{Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\phantom{6x + } 13y + 19z =& 4
&& \text{Add}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

y + 6z =& 3
&& \text{Equation 4}
\\
13y + 19z =& 4
&& \text{Equation 5}

\end{aligned}
\end{equation}
$


We write the equations in two variables as a system


$
\begin{equation}
\begin{aligned}

-13y - 78z =& -39
&& -13 \times \text{ Equation 4}
\\
13y + 19z =& 4
&&
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\phantom{13y } -59z =& -35
&& \text{Add}
\\
z =& \frac{35}{59}
&& \text{Divide each side by $-59$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

y + 6 \left( \frac{35}{59} \right) =& 3
&& \text{Substitute } z = \frac{35}{59} \text{ in Equation 4}
\\
\\
y + \frac{210}{59} =& 3
&& \text{Multiply}
\\
\\
y =& - \frac{33}{59}
&& \text{Subtract each side by } \frac{210}{59}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

x + 2 \left( - \frac{33}{59} \right) + 3 \left( \frac{35}{59} \right) =& 1
&& \text{Substitute } y = \frac{33}{59} \text{ and } z = \frac{35}{59}
\\
\\
x - \frac{66}{59} + \frac{105}{59} =& 1
&& \text{Multiply}
\\
\\
x + \frac{39}{59} =& 1
&& \text{Combine like terms}
\\
\\
x =& \frac{20}{59}
&& \text{Subtract each side by } \frac{171}{59}

\end{aligned}
\end{equation}
$



The ordered triple is $\displaystyle \left( \frac{20}{59}, - \frac{33}{59}, \frac{35}{59} \right)$.