We are given the system of equations:

1. $2x - 3y = 1$
2. $4x - 2y = 6$

Step 1: Solve using substitution or elimination

We'll use the elimination method. First, we'll multiply the first equation by 2 to align the coefficients of $x$.

$2(2x - 3y) = 2(1) \implies 4x - 6y = 2$

Now, we subtract this from the second equation:

$(4x - 2y) - (4x - 6y) = 6 - 2$

This simplifies to:

$4y = 4 \implies y = 1$

Step 2: Substitute $y = 1$ into one of the original equations

We'll substitute $y = 1$ into the first equation:

$2x - 3(1) = 1 \implies 2x - 3 = 1 \implies 2x = 4 \implies x = 2$

Solution:

The solution is $(x, y) = (2, 1)$.

Since this doesn't match any of the provided options, there may be a misprint in the choices. However, based on the solution, the answer is $(2, 1)$.

Would you like further details or have any questions? Here are 5 related problems for practice:

1. Solve the system of equations: $x + y = 4$ and $2x - y = 3$.
2. Solve the system: $3x - y = 7$ and $2x + y = 4$.
3. What is the solution to $5x + 2y = 12$ and $3x - 4y = -2$?
4. Find the values of $x$ and $y$ for the equations $6x - y = 5$ and $4x + y = 3$.
5. Solve $3x + 4y = 10$ and $2x - 5y = 1$.

Tip: Always check the solution by substituting the values of $x$ and $y$ back into the original equations.