# Composition of systems of equations

Suppose $$2x + 3y = u$$ $$x - 4y = v$$

and further that $$3u - 5v = c$$ $$2u + 3v = d$$

Express c and d in terms of $x$ and $y$ by matrix multiplication.

It's quite easy by direct substitution but I can;t work out how to use matrix multiplication. Any ideas? Thanks in advance!

\begin{align*} 2x + 3y& = u\\ x - 4y &= v \end{align*}\quad \underset{\substack{\text{convert to}\\ \text{matrix language}}}{\leadsto}\quad \begin{bmatrix} 2 & \hphantom{-}3\\ 1 &-4\end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix}u\\ v\end{bmatrix} \begin{align*} 3u - 5v &= c\\ 2u + 3v &= d \end{align*}\quad \underset{\substack{\text{convert to}\\ \text{matrix language}}}{\leadsto}\quad \begin{bmatrix} 3 & -5\\ 2 &\hphantom{-}3\end{bmatrix}\begin{bmatrix} u\\ v\end{bmatrix}=\begin{bmatrix}c\\ d\end{bmatrix}
\begin{bmatrix} 3 & -5\\ 2 &\hphantom{-}3\end{bmatrix}\begin{bmatrix} u\\ v\end{bmatrix}=\begin{bmatrix} 3 & -5\\ 2 &\hphantom{-}3\end{bmatrix}\Bigg(\begin{bmatrix} 2 & \hphantom{-}3\\ 1 &-4\end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}\Bigg)=\begin{bmatrix}c\\ d\end{bmatrix}\implies\begin{bmatrix} p & q\\ r &s\end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix}c\\ d\end{bmatrix}\\[0.4in] \quad\text{ where }\quad \begin{bmatrix} p & q\\ r &s\end{bmatrix}=\begin{bmatrix} 3 & -5\\ 2 &\hphantom{-}3\end{bmatrix}\begin{bmatrix} 2 & \hphantom{-}3\\ 1 &-4\end{bmatrix}.\\[0.4in] \begin{bmatrix} p & q\\ r &s\end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix}c\\ d\end{bmatrix}\quad\underset{\substack{\text{convert to}\\ \text{equation language}}}{\leadsto}\quad \begin{align*} px + qy& = c\\ rx + sy &= d \end{align*}