The system of equations provided in the image is:

1. $-4x + 9y = 9$
2. $x - 3y = -6$

Step-by-step Solution:

1. Solve the second equation for $x$:

From the equation $x - 3y = -6$, solve for $x$:

$x = 3y - 6$

2. Substitute $x = 3y - 6$ into the first equation:

Substitute into $-4x + 9y = 9$:

$-4(3y - 6) + 9y = 9$

Simplify:

$-12y + 24 + 9y = 9$

Combine like terms:

$-3y + 24 = 9$

Subtract 24 from both sides:

$-3y = 9 - 24$

$-3y = -15$

Divide by -3:

$y = 5$

3. Substitute $y = 5$ into the equation for $x$:

Substitute into $x = 3y - 6$:

$x = 3(5) - 6$

$x = 15 - 6 = 9$

Final Solution:

$x = 9, \quad y = 5$

Would you like any more details on this? Here are some related questions you might find useful:

1. What happens if we graph the system of equations?
2. How do we solve this system using matrices or determinants?
3. Can this system be solved by substitution or elimination? Which is faster?
4. What are some methods for solving systems of equations with more than two variables?
5. How can solving systems of equations be applied in real-life situations?

Tip: When solving systems of equations, try to choose the method (substitution, elimination, or graphing) that fits the structure of the problem for quicker results.