Answer
The ${{D}_{x}}\text{ and }{{D}_{y}}$ are represented in terms of the coefficients and constants.
Work Step by Step
Given a linear system in two variables,
$\begin{align}
& {{a}_{1}}x+{{b}_{1}}y={{c}_{1}} \\
& {{a}_{2}}x+{{b}_{2}}y={{c}_{2}} \\
\end{align}$
Where $a_1$, $a_2$, $b_1$, $b_2$ are coefficients and $c_1$ and $c_2$ are constants.
Then,
$x=\frac{\left| \begin{matrix}
{{c}_{1}} & {{b}_{1}} \\
{{c}_{2}} & {{b}_{2}} \\
\end{matrix} \right|}{\left| \begin{matrix}
{{a}_{1}} & {{b}_{1}} \\
{{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right|}$
And
$y=\frac{\left| \begin{matrix}
{{a}_{1}} & {{c}_{1}} \\
{{a}_{2}} & {{c}_{2}} \\
\end{matrix} \right|}{\left| \begin{matrix}
{{a}_{1}} & {{b}_{1}} \\
{{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right|}\text{ }$
Where,
$\left| \begin{matrix}
{{a}_{1}} & {{b}_{1}} \\
{{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right|\ne 0$
Or
$x=\frac{{{D}_{x}}}{D}\text{ and }y=\frac{{{D}_{y}}}{D}$
Where,
$\begin{align}
& {{D}_{x}}=\left| \begin{matrix}
{{c}_{1}} & {{b}_{1}} \\
{{c}_{2}} & {{b}_{2}} \\
\end{matrix} \right| \\
& {{D}_{y}}=\left| \begin{matrix}
{{a}_{1}} & {{c}_{1}} \\
{{a}_{2}} & {{c}_{2}} \\
\end{matrix} \right| \\
& \text{ and } \\
& {{D}_{{}}}=\left| \begin{matrix}
{{a}_{1}} & {{b}_{1}} \\
{{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right|\ne 0 \\
\end{align}$
So, ${{D}_{x}}\text{ and }{{D}_{y}}$ are represented in terms of the coefficients and constants.
