Let's solve the system of equations using the elimination method.

We have the system:

1. $-3x + y = -14$
2. $2x - y = 7$

Step 1: Add the two equations

The goal is to eliminate one of the variables by adding or subtracting the equations. Notice that the $y$-terms are opposite in sign (one is $+y$ and the other is $-y$), so if we add the two equations, the $y$-terms will cancel out.

Add the two equations:

$(-3x + y) + (2x - y) = -14 + 7$

This simplifies to:

$-3x + 2x + y - y = -7$ $-x = -7$

Step 2: Solve for $x$

Now, solve for $x$:

$x = 7$

Step 3: Substitute $x = 7$ into one of the original equations

We can substitute $x = 7$ into either equation. Let's use the first equation:

$-3x + y = -14$ $-3(7) + y = -14$ $-21 + y = -14$

Now, solve for $y$:

$y = -14 + 21$ $y = 7$

Final Solution:

The solution to the system is:

$x = 7, \quad y = 7$

Would you like any further details or clarifications on this? Here are some additional questions you can explore:

1. How do you know when to use elimination versus substitution?
2. What if both variables had the same coefficient in front of them?
3. Can the elimination method be used for systems of three equations?
4. How would this problem look if the equations were in a different format (e.g., fractions)?
5. What are some common mistakes made during the elimination process?

Tip: Always check your solution by substituting the values of $x$ and $y$ back into the original equations to verify they satisfy both equations!