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We are tasked with solving the system of equations using the method of elimination. The system is:

\[
\begin{aligned}
1. \ \ x - y &= -1 \\
2. \ \ -2x + 3y &= -5
\end{aligned}
\]

### Step 1: Eliminate one variable
We can eliminate the variable \(x\) by multiplying the first equation by 2, so the coefficients of \(x\) in both equations are opposites:

\[
2(x - y) = 2(-1) \quad \Rightarrow \quad 2x - 2y = -2
\]

Now, we have the system:

\[
\begin{aligned}
2x - 2y &= -2 \\
-2x + 3y &= -5
\end{aligned}
\]

### Step 2: Add the equations
Add the two equations to eliminate \(x\):

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

Simplifying:

\[
0x + y = -7
\]

So, we get:

\[
y = -7
\]

### Step 3: Substitute the value of \(y\)
Now that we have \(y = -7\), substitute this into the first original equation to find \(x\):

\[
x - (-7) = -1 \quad \Rightarrow \quad x + 7 = -1
\]

Solving for \(x\):

\[
x = -1 - 7 = -8
\]

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

\[
x = -8, \quad y = -7
\]

Would you like more details or have any questions?

Here are 5 related questions you might want to consider:
1. How do you solve systems of equations by substitution instead of elimination?
2. Can elimination always be used to solve any system of linear equations?
3. What is the difference between consistent and inconsistent systems?
4. How do you solve a system of equations with three variables using elimination?
5. Can elimination be applied to non-linear systems of equations?

**Tip:** Always check your solution by substituting both values back into the original equations to ensure accuracy.

Solve the system of equations using the method of elimination:
1. x - y = -1
2. -2x + 3y = -5

Learn how to solve a system of linear equations using the method of elimination. This step-by-step guide walks through solving the system of equations x - y = -1 and -2x + 3y = -5. This method is suitable for students in Grades 8-10 and covers key algebraic concepts.

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