To solve the system of equations and find the value of \( y \), we can use either substitution or elimination. Let's use the **elimination method** for this problem:

The system of equations is:

1. \( 2x + 3y = 12 \)
2. \( 4x + 7y + 2 = 0 \) (which can be rewritten as \( 4x + 7y = -2 \))

### Step 1: Eliminate one variable
To eliminate \( x \), we multiply the first equation by 2, so both equations have the same coefficient for \( x \):

- Multiply the first equation \( 2x + 3y = 12 \) by 2:
  \[
  4x + 6y = 24
  \]

Now, subtract the second equation \( 4x + 7y = -2 \) from this result:

\[
(4x + 6y) - (4x + 7y) = 24 - (-2)
\]
\[
4x + 6y - 4x - 7y = 24 + 2
\]
\[
- y = 26
\]

### Step 2: Solve for \( y \)
\[
y = -26
\]

Thus, the value of \( y \) is \( -26 \).

Would you like to go through the steps for finding \( x \) as well or need further details?

### 5 Related Questions:
1. What is the value of \( x \) in this system?
2. How can we solve this system using the substitution method?
3. How can this method be applied to larger systems of equations?
4. How does the elimination method compare to substitution in terms of efficiency?
5. Can this system be solved graphically? If so, what would the lines look like?

**Tip**: When dealing with linear equations, elimination works best when coefficients of one variable are easy to equalize, while substitution is effective when one variable is easily isolated.

This page explains how to solve the system of equations 2x + 3y = 12 and 4x + 7y = -2 using the elimination method. It provides a step-by-step guide, solving for y as -26. Suitable for students in Grades 8-10 learning about algebra and systems of linear equations.

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