We are given the system of equations:

1. \( 2x + y = 3 \)
2. \( -2x + 5y = -9 \)

Let's solve this system using the **elimination method**.

### Step 1: Add the two equations
Since the coefficients of \( x \) are opposites ( \( 2x \) and \( -2x \) ), we can add the two equations to eliminate \( x \).

\[
(2x + y) + (-2x + 5y) = 3 + (-9)
\]
\[
0x + 6y = -6
\]
\[
6y = -6
\]
\[
y = \frac{-6}{6} = -1
\]

### Step 2: Substitute \( y = -1 \) into one of the original equations
We'll substitute it into the first equation, \( 2x + y = 3 \).

\[
2x + (-1) = 3
\]
\[
2x - 1 = 3
\]
\[
2x = 3 + 1 = 4
\]
\[
x = \frac{4}{2} = 2
\]

### Final Solution:
The solution to the system is:
\[
x = 2, \quad y = -1
\]

Let me know if you'd like more details or further explanation on any step!

### Related Questions:
1. How would the solution change if one of the coefficients in the second equation were different?
2. Can you solve this system using the substitution method?
3. What happens if you multiply both equations by a constant before solving?
4. How can you graphically interpret the solution to this system?
5. How would this process differ for a system with no solution or infinite solutions?

**Tip**: Elimination is a powerful method when one of the variables already has opposite coefficients!

Explore how to solve the system of linear equations 2x + y = 3 and -2x + 5y = -9 using the elimination method. This step-by-step solution covers key algebraic concepts, making it ideal for students in Grades 8-10.

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