https://file.aiquickdraw.com/mathbot/file/17248658862111o13zell.jpg

The image shows a system of linear equations:

\[
\begin{cases}
2x - y = -3 \\
-5x + y = 9
\end{cases}
\]

To solve this system, we'll use the **elimination method** again.

### Step 1: Add the two equations
First, notice that adding the two equations will allow us to eliminate \( y \):

\[
(2x - y) + (-5x + y) = -3 + 9
\]

This simplifies to:

\[
2x - 5x = 6
\]

\[
-3x = 6
\]

### Step 2: Solve for \( x \)
Now, divide by \(-3\):

\[
x = \frac{6}{-3} = -2
\]

### Step 3: Substitute \( x = -2 \) into one of the original equations
Substitute \( x = -2 \) into the first equation:

\[
2(-2) - y = -3
\]

\[
-4 - y = -3
\]

### Step 4: Solve for \( y \)
Add 4 to both sides:

\[
-y = -3 + 4
\]

\[
-y = 1
\]

Multiply by \(-1\):

\[
y = -1
\]

### Solution
The solution to the system of equations is:

\[
\boxed{(-2, -1)}
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How does the solution relate to the point of intersection of the two lines represented by the equations?
2. What would the solution be if the second equation had a different constant on the right-hand side?
3. Can you confirm the solution by graphing the equations?
4. How would this problem change if you used the substitution method instead of elimination?
5. What happens if you multiply one of the equations by a constant before adding or subtracting?

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

Learn how to solve a system of linear equations using the elimination method with a detailed step-by-step solution. This problem covers key mathematical concepts such as systems of linear equations and the elimination method. Suitable for students in Grades 9-12.

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