https://file.aiquickdraw.com/mathbot/file/1728623943428b7fihylt.jpg

The problem in the image asks to solve the system of linear equations using the addition method:

\[
\begin{aligned}
-9x - 7y &= 57 \\
-4x + 5y &= 31
\end{aligned}
\]

### Steps to Solve Using the Addition Method:

1. **Eliminate one variable**: We want to eliminate either \(x\) or \(y\). Let's eliminate \(x\) by making the coefficients of \(x\) in both equations equal. The least common multiple of \(-9\) and \(-4\) is 36, so we multiply the first equation by 4 and the second equation by 9:
   
   \[
   4(-9x - 7y) = 4(57) \implies -36x - 28y = 228
   \]
   \[
   9(-4x + 5y) = 9(31) \implies -36x + 45y = 279
   \]

2. **Add the equations** to eliminate \(x\):
   
   \[
   (-36x - 28y) + (-36x + 45y) = 228 + 279
   \]
   Simplifying:
   \[
   -36x + 36x + (-28y + 45y) = 507 \implies 17y = 507
   \]

3. **Solve for \(y\)**:
   \[
   y = \frac{507}{17} = 29.8
   \]

So \(y = 29.8\).

4. **Substitute \(y = 29.8\)** back into one of the original equations to find \(x\). Let's use the second equation:
   
   \[
   -4x + 5(29.8) = 31
   \]
   Simplifying:
   \[
   -4x + 149 = 31
   \]
   \[
   -4x = 31 - 149 \implies -4x = -118
   \]
   \[
   x = \frac{-118}{-4} = 29.5
   \]

Thus, \(x = 29.5\) and \(y = 29.8\).

### Final Solution:
The system has one solution, which is \(x = 29.5\) and \(y = 29.8\).

Would you like further clarification on any step? 

Here are 5 related questions to practice:
1. Can you solve a similar system using substitution instead of addition?
2. How do we handle a system where the variables cancel out completely?
3. What are the differences between the elimination method and substitution method?
4. How do you verify the solution to a system of equations?
5. What happens if the two equations represent parallel lines?

**Tip:** Always check your final solution by plugging both values of \(x\) and \(y\) into the original equations to ensure consistency.

Solve the system with the addition method:

-9x - 7y = 57
-4x + 5y = 31

This problem demonstrates solving a system of linear equations using the addition (elimination) method. It involves algebraic manipulation to eliminate a variable and find the solution for x and y. Suitable for students in Grades 8-10.

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