Intermediate Algebra, Chapter 4, 4.2, Section 4.2, Problem 18

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

6.2x - 1.4 y + 2.4z =& -1.80 \\
3.1x + 2.8y - 0.2z =& 5.68 \\
9.3x - 8.4y - 4.8z =& -34.20

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


$
\begin{equation}
\begin{aligned}

6.2x - 1.4y + 2.4z =& -1.80
&& \text{Equation 1}
\\
37.2x + 33.6y - 2.4z =& 68.16
&& 12 \times \text{ Equation 2}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

43.4x + 32.2y \phantom{-2.4z} =& 66.36
&& \text{Add}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

12.4x - 2.8y + 4.8z =& -3.60
&& 2 \times \text{ Equation 1}
\\
9.3x - 8.4y - 4.8z =& -34.20
&& \text{Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

21.7x - 11.2y \phantom{-4.8z} =& -37.80
&& \text{Add}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

43.4x + 32.2y =& 66.36
&& \text{Equation 4}
\\
21.7x - 11.2y =& -37.80
&& \text{Equation 5}

\end{aligned}
\end{equation}
$


We write the equations in two variables as a system


$
\begin{equation}
\begin{aligned}

43.4x + 32.2y =& 66.36
&& \text{Equation 4}
\\
62.3875x - 32.2y =& -108.675
&& 2.875 \times \text{ Equation 5}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

105.7875x \phantom{-32.2y} =& -42.315
&& \text{Add}
\\
x =& -0.4
&& \text{Divide each side by $105.7875$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

43.4(-0.4) + 32.2y =& 66.36
&& \text{Substitute } x = -0.4 \text{ in Equation 4}
\\
-17.36 + 32.2y =& 66.36
&& \text{Multiply}
\\
32.2y =& 83.72
&& \text{Add each side by $17.36$}
\\
y =& 2.6
&& \text{Divide each side by $32.2$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

6.2(-0.4) - 1.4(2.6) + 2.4z =& -1.8
&& \text{Substitute } x = -0.4 \text{ and } y = 2.6 \text{ in Equation 1}
\\
-2.48 - 3.64 + 2.4z =& -1.8
&& \text{Multiply}
\\
-6.12 + 2.4z =& -1.8
&& \text{Combine like terms}
\\
2.4z =& 4.32
&& \text{Add each side by $6.12$}
\\
z =& 1.8
&& \text{Divide each side by $2.4$}

\end{aligned}
\end{equation}
$



The ordered triple is $\displaystyle \left( -0.4,2.6,1.8 \right)$.