1. **State the problem:** We have two matrix transformations: matrix $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ transforms point $A$ to point $B$, and matrix $\begin{pmatrix}3 & 1 \\ 0 & 1\end{pmatrix}$ transforms point $B$ to point $C$ with coordinates $(4,7)$. We need to find the coordinates of $A$. 2. **Write the equations for the transformations:** Let $A = \begin{pmatrix}a_1 \\ a_2\end{pmatrix}$, $B = \begin{pmatrix}b_1 \\ b_2\end{pmatrix}$, and $C = \begin{pmatrix}4 \\ 7\end{pmatrix}$. From the first transformation: $$B = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} A = \begin{pmatrix}0 \cdot a_1 + 1 \cdot a_2 \\ 1 \cdot a_1 + 0 \cdot a_2\end{pmatrix} = \begin{pmatrix}a_2 \\ a_1\end{pmatrix}$$ 3. **From the second transformation:** $$C = \begin{pmatrix}3 & 1 \\ 0 & 1\end{pmatrix} B = \begin{pmatrix}3b_1 + b_2 \\ 0 \cdot b_1 + 1 \cdot b_2\end{pmatrix} = \begin{pmatrix}4 \\ 7\end{pmatrix}$$ This gives the system: $$3b_1 + b_2 = 4$$ $$b_2 = 7$$ 4. **Solve for $b_1$ and $b_2$:** Substitute $b_2 = 7$ into the first equation: $$3b_1 + 7 = 4$$ $$3b_1 = 4 - 7 = -3$$ $$b_1 = \frac{-3}{3} = -1$$ So, $$B = \begin{pmatrix}-1 \\ 7\end{pmatrix}$$ 5. **Recall that $B = \begin{pmatrix}a_2 \\ a_1\end{pmatrix}$, so:** $$a_2 = -1$$ $$a_1 = 7$$ 6. **Therefore, the coordinates of $A$ are:** $$A = \begin{pmatrix}7 \\ -1\end{pmatrix}$$