We are given the system of linear equations:

1. $-5x + 2y = -11$
2. $-x + 2y = -7$

We can solve this system using the elimination method or the substitution method. I'll use the elimination method here, focusing on eliminating one variable to solve for the other.

Step 1: Subtract the second equation from the first equation

Since both equations have the same coefficient for $y$ (i.e., $2y$), we can subtract the second equation from the first to eliminate $y$.

$(-5x + 2y) - (-x + 2y) = -11 - (-7)$

Simplifying:

$-5x + 2y + x - 2y = -11 + 7$

This simplifies to:

$-4x = -4$

Step 2: Solve for $x$

Now, divide both sides of the equation by $-4$:

$x = 1$

Step 3: Substitute the value of $x$ into one of the original equations

Substitute $x = 1$ into the second equation:

$-x + 2y = -7$

Substitute $x = 1$:

$-(1) + 2y = -7$

Simplifying:

$-1 + 2y = -7$

Add 1 to both sides:

$2y = -6$

Now, divide by 2:

$y = -3$

Final Solution:

The solution to the system of equations is:

$x = 1, \quad y = -3$

Let me know if you need further details or clarifications.

Related Questions:

1. How would you solve the system using the substitution method?
2. Can the system be solved graphically? If so, what would the intersection point be?
3. What happens if both equations represent the same line? How do we handle that?
4. Can we apply matrix methods (like Gauss-Jordan elimination) to solve this system?
5. How do we determine if a system of equations has no solution or infinitely many solutions?

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