How to solve a system of 3 equations with Cramer's Rule? I am given the following system of 3 simultaneous equations:
$$
\begin{align*}
4a+c &= 4\\
19a + b - 3c &= 3\\
7a + b &= 1\end{align*}
$$
How do I solve using Cramers' rule?
For one, I do know that by putting as a matrix the LHS
$$\begin{pmatrix}
4&0&1\\19&1&-3\\7&1&0
\end{pmatrix}$$
and then computing its determinant to be $24$ is of some use...but could somebody give more ideas?

$$\begin{pmatrix}
4&0&1\\19&1&-3\\7&1&0
\end{pmatrix}=\frac{1}{a}\begin{pmatrix}
4a&0&1\\19a&1&-3\\7a&1&0
\end{pmatrix}$$
$$\Rightarrow \frac{1}{a}\begin{pmatrix}
(4a+0.b+1.c)&0&1\\(19a+b-3c)&1&-3\\(7a+b+0.c)&1&0
\end{pmatrix}$$
$$\Rightarrow \frac{1}{a}\begin{pmatrix}
4&0&1\\3&1&-3\\1&1&0
\end{pmatrix}$$
$$\Rightarrow \begin{pmatrix}
4&0&1\\19&1&-3\\7&1&0
\end{pmatrix}=\frac{1}{a}\begin{pmatrix}
4&0&1\\3&1&-3\\1&1&0
\end{pmatrix}$$
$$\Rightarrow \frac{det(\begin{pmatrix}
4&0&1\\3&1&-3\\1&1&0
\end{pmatrix})}{det \begin{pmatrix}
4&0&1\\19&1&-3\\7&1&0
\end{pmatrix} }=a$$
Similarly for others.
