1. **State the problem:** Given the equation $$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{5} \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$ find the value of $y$. 2. **Perform matrix multiplication:** Multiply the 2x2 matrix by the 2x1 vector: $$\begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 1 \times 3 + (-2) \times 4 \\ 2 \times 3 + 1 \times 4 \end{bmatrix} = \begin{bmatrix} 3 - 8 \\ 6 + 4 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}$$ 3. **Multiply by the scalar $\frac{1}{5}$:** Multiply each element by $\frac{1}{5}$: $$\frac{1}{5} \begin{bmatrix} -5 \\ 10 \end{bmatrix} = \begin{bmatrix} \frac{-5}{5} \\ \frac{10}{5} \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}$$ 4. **Extract the value of $y$:** The vector is now $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}$, so the value of $y$ is $2$. **Final answer:** $$y = 2$$