https://file.aiquickdraw.com/mathbot/file/1729696685324wp9ru5pb.jpg

The system of linear equations shown in the image is:

\[
\begin{cases}
-7x + 3y = -65 \\
-7x + 10y = -135
\end{cases}
\]

We can solve this system using the elimination method by eliminating \(x\). Since both equations have the same coefficient for \(x\), subtracting one equation from the other will eliminate \(x\) and allow us to solve for \(y\).

### Step 1: Subtract the two equations
Subtract the first equation from the second equation:

\[
(-7x + 10y) - (-7x + 3y) = -135 - (-65)
\]

This simplifies to:

\[
7y = -70
\]

### Step 2: Solve for \(y\)

\[
y = \frac{-70}{7} = -10
\]

### Step 3: Substitute \(y = -10\) into one of the original equations

Substitute \(y = -10\) into the first equation:

\[
-7x + 3(-10) = -65
\]

Simplifying this:

\[
-7x - 30 = -65
\]

Add 30 to both sides:

\[
-7x = -35
\]

Solve for \(x\):

\[
x = \frac{-35}{-7} = 5
\]

### Final Solution
The solution to the system of equations is:
\[
x = 5, \quad y = -10
\]

Would you like more details or have any questions?

### Related Questions:
1. What are other methods to solve this system, such as substitution or matrix methods?
2. How can systems of equations be applied to real-world problems?
3. How do we know if a system of equations has no solution or infinitely many solutions?
4. What happens if the equations had coefficients that were multiples of each other?
5. Can this system be solved graphically, and what would the graph look like?

### Tip:
Always check your solution by substituting the values of \(x\) and \(y\) back into both original equations to ensure correctness.

Solve the system of equations: -7x + 3y = -65 and -7x + 10y = -135

This solution explains how to solve a system of linear equations (-7x + 3y = -65 and -7x + 10y = -135) using the elimination method. Follow a step-by-step approach suitable for students in Grades 9-11.

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