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To solve the system of equations using the elimination method, we'll eliminate one of the variables by adding or subtracting the equations. The given system is:

\[
\begin{aligned}
4x + 2y &= -6 \quad \text{(Equation 1)} \\
-3x - 2y &= 7 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Add the equations to eliminate \(y\).
Since the coefficients of \(y\) are \(+2\) and \(-2\), adding the two equations will eliminate \(y\):

\[
(4x + 2y) + (-3x - 2y) = -6 + 7
\]

Simplifying this:

\[
4x - 3x + 2y - 2y = 1
\]

\[
x = 1
\]

### Step 2: Substitute \(x = 1\) into one of the original equations to find \(y\).
We can use Equation 1:

\[
4(1) + 2y = -6
\]

\[
4 + 2y = -6
\]

\[
2y = -6 - 4
\]

\[
2y = -10
\]

\[
y = -5
\]

### Final Answer:
The solution to the system of equations is \((x, y) = (1, -5)\).

Would you like more details, or do you have any questions?

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Here are five related questions:
1. What is the substitution method, and how does it differ from the elimination method?
2. Can the elimination method be used when the coefficients of the variables don't directly cancel out? How?
3. How would the solution change if the system of equations was dependent?
4. What are the possible outcomes (solution types) for a system of linear equations?
5. How can you verify the solution of a system of equations?

**Tip:** Always double-check your final solution by substituting the values back into the original equations.

Solve this system of equations by using the elimination method:
4x + 2y = -6
-3x - 2y = 7

Learn how to solve the system of equations 4x + 2y = -6 and -3x - 2y = 7 using the elimination method. This problem covers key algebraic concepts, including solving linear equations by eliminating variables. Suitable for students in Grades 8-10.

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