We can solve this system of equations using substitution or elimination. Let's use substitution to find the value of $x$.

The system of equations is:

1. $-2x + y = 6$
2. $x - y = 1$

Step 1: Solve the second equation for $y$:

From the second equation: $x - y = 1$ Solve for $y$: $y = x - 1$

Step 2: Substitute $y = x - 1$ into the first equation:

Substitute this into the first equation: $-2x + (x - 1) = 6$ Simplify the equation: $-2x + x - 1 = 6$ $-x - 1 = 6$

Step 3: Solve for $x$:

Add 1 to both sides: $-x = 7$ Multiply both sides by -1: $x = -7$

Thus, the value of $x$ is $\boxed{-7}$.

Would you like more details on this method? Here are some related questions:

1. How can we solve the system using the elimination method?
2. What is the value of $y$ in this system?
3. How can systems of equations be used in real-world scenarios?
4. What are some other methods for solving systems of equations?
5. What happens when a system has no solutions or infinitely many solutions?

Tip: Always check your solution by substituting the values back into the original equations!