To solve the given system of equations using substitution:

System of equations:

1. $-2x + 3y = -12$
2. $x - 2y = 7$

Steps to solve:

1. Solve one equation for one variable:
   From the second equation $x - 2y = 7$, solve for $x$: $x = 2y + 7$

2. Substitute $x = 2y + 7$ into the first equation:
   Replace $x$ in $-2x + 3y = -12$: $-2(2y + 7) + 3y = -12$

3. Simplify and solve for $y$:
   Distribute $-2$: $-4y - 14 + 3y = -12$ Combine like terms: $-y - 14 = -12$ Solve for $y$: $-y = 2 \quad \Rightarrow \quad y = -2$

4. Substitute $y = -2$ back into $x = 2y + 7$:
   Replace $y$: $x = 2(-2) + 7$ Simplify: $x = -4 + 7 \quad \Rightarrow \quad x = 3$

Final Solution:

$x = 3, \, y = -2$

Let me know if you'd like further clarification or more details!

5 Follow-Up Questions:

1. How can substitution be used in solving systems with more than two variables?
2. What is the advantage of using substitution over elimination in solving systems of equations?
3. How would the solution change if one equation was nonlinear (e.g., $x^2 - 2y = 7$)?
4. What happens if the system has no solution or infinitely many solutions? How do you identify those cases?
5. Can you graphically interpret the solution of this system?

One Tip:

Always double-check your substitution and calculations to ensure accuracy, especially with signs and operations!