Solve the system of linear equations using Cramer's Rule: 2x + y = 5, x − y = 1

The coefficient matrix is formed by the coefficients of $x$ and $y$:

$$D = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}$$

The determinant of $D$ is:

$$\det(D) = (2)(-1) - (1)(1) = -2 - 1 = -3$$

$D_x$ is formed by replacing the first column of $D$ with the constant terms (5 and 1):

$$D_x = \begin{bmatrix} 5 & 1 \\ 1 & -1 \end{bmatrix}$$

The determinant of $D_x$ is:

$$\det(D_x) = (5)(-1) - (1)(1) = -5 - 1 = -6$$

$D_y$ is formed by replacing the second column of $D$ with the constant terms (5 and 1):

$$D_y = \begin{bmatrix} 2 & 5 \\ 1 & 1 \end{bmatrix}$$

The determinant of $D_y$ is:

$$\det(D_y) = (2)(1) - (5)(1) = 2 - 5 = -3$$

Cramer's Rule states that $x = \frac{\det(D_x)}{\det(D)}$ and $y = \frac{\det(D_y)}{\det(D)}$.

$$x = \frac{-6}{-3} = 2$$ $$y = \frac{-3}{-3} = 1$$

So, the solution is $x=2$ and $y=1$.

Comparing this with the given options:
a. $x=2, y=1$
b. $x=3, y=-1$
c. $x=1, y=2$
d. $x=2, y=1$

The correct option is a.

The final answer is $\boxed{\text{a. } x=2, y=1}$.