Determine the solution of the system of equations: $\begin{array}{l}2 x-3 y=1 \\ x-y=1\end{array}$

Step 1 ：We have the system of equations: \(\begin{array}{l}2 x-3 y=1 \\ x-y=1\end{array}\)

Step 2 ：Let's solve the second equation for x: \(x = y + 1\)

Step 3 ：Substitute x in the first equation with the expression we got from the second equation: \(2*(y + 1) - 3*y = 1\)

Step 4 ：Solve for y: \(y = 1\)

Step 5 ：Substitute y = 1 back into the second equation to solve for x: \(x = 1 + 1\)

Step 6 ：Solve for x: \(x = 2\)

Step 7 ：Final Answer: The solution to the system of equations is \(\boxed{x = 2, y = 1}\)