To solve the system of linear equations:

\[
\begin{aligned}
-2x - 3y &= -7 \quad \text{(Equation 1)} \\
-3x + 2y &= -17 \quad \text{(Equation 2)}
\end{aligned}
\]

We can use either the substitution method or the elimination method. Here, I'll use the elimination method.

### Step 1: Multiply each equation to eliminate one variable
Let's eliminate \( y \). Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of \( y \) the same:

\[
\begin{aligned}
2(-2x - 3y) &= 2(-7) \\
3(-3x + 2y) &= 3(-17)
\end{aligned}
\]

This results in:

\[
\begin{aligned}
-4x - 6y &= -14 \quad \text{(Equation 3)} \\
-9x + 6y &= -51 \quad \text{(Equation 4)}
\end{aligned}
\]

### Step 2: Add the equations to eliminate \( y \)
Add Equation 3 and Equation 4:

\[
\begin{aligned}
(-4x - 6y) + (-9x + 6y) &= -14 + (-51) \\
-13x &= -65
\end{aligned}
\]

### Step 3: Solve for \( x \)
Divide both sides by \(-13\):

\[
x = \frac{-65}{-13} = 5
\]

### Step 4: Substitute \( x = 5 \) back into one of the original equations
Substitute \( x = 5 \) into Equation 1:

\[
-2(5) - 3y = -7
\]

Simplify and solve for \( y \):

\[
-10 - 3y = -7
\]

Add 10 to both sides:

\[
-3y = 3
\]

Divide by \(-3\):

\[
y = -1
\]

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

\[
x = 5, \quad y = -1
\]

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. How can you verify the solution to a system of equations?
2. What is the substitution method for solving a system of equations?
3. How does the graph of these equations look?
4. Can a system of linear equations have no solution? Why?
5. What are the conditions for a system of equations to have infinite solutions?

**Tip:** When using the elimination method, always check if multiplying one or both equations by a constant can simplify the process of eliminating a variable.

Learn how to solve the system of linear equations -2x - 3y = -7 and -3x + 2y = -17 using the elimination method. Detailed step-by-step solution provided. Suitable for high school students.

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