1. **State the problem:** We have the matrix equation $$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{5} \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$ and we need to find the value of $x$. 2. **Multiply the 2x2 matrix by the 2x1 matrix:** Calculate each element: $$\begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} (1)(3) + (-2)(4) \\ (2)(3) + (1)(4) \end{bmatrix} = \begin{bmatrix} 3 - 8 \\ 6 + 4 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}$$ 3. **Multiply the result 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. **Identify $x$ and $y$ from the resulting vector:** $$x = -1, \quad y = 2$$ 5. **Answer:** The value of $x$ is $\boxed{-1}$, which corresponds to choice d.