<s>[INST] Solving for three unknowns given three linear equations Have to admit, I forgot how to do simple 9th grade algebra problems. I need to know how to do this stuff for an exam tomorrow.  I am doing recurrences and the only thing I need to do is find $a, b$, and $c$ from the following
$$\begin{align*}
a + b + c &= 3\\
4a + 2b+ c&= 6\\
9a + 3b + c& = 13
\end{align*}$$
Can someone refresh my memory on how to solve for  $a, b$ and $c$
?
 [/INST] You could represent your system as a single equation using matrices:
$$
\left[
\begin{matrix}
1 & 1 & 1\\
4 & 2 & 1\\
9 & 3 & 1
\end{matrix}
\right]
\left[
\begin{matrix}
a\\
b\\
c
\end{matrix}
\right]
=
\left[
\begin{matrix}
3\\
6\\
13
\end{matrix}
\right]
$$
To isolate the matrix of unknowns, multiply on the left by the inverse of the matrix of coefficients:
$$
\left[
\begin{matrix}
1 & 1 & 1\\
4 & 2 & 1\\
9 & 3 & 1
\end{matrix}
\right]^{-1}
\left[
\begin{matrix}
1 & 1 & 1\\
4 & 2 & 1\\
9 & 3 & 1
\end{matrix}
\right]
\left[
\begin{matrix}
a\\
b\\
c
\end{matrix}
\right]
=
\left[
\begin{matrix}
1 & 1 & 1\\
4 & 2 & 1\\
9 & 3 & 1
\end{matrix}
\right]^{-1}
\left[
\begin{matrix}
3\\
6\\
13
\end{matrix}
\right]
$$
which simplifies to
$$
\left[
\begin{matrix}
a\\
b\\
c
\end{matrix}
\right]
=
\left[
\begin{matrix}
2\\
-3\\
4
\end{matrix}
\right]
$$
That is, $a = 2$, $b = -3$, and $c = 4$.
</s>