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We are asked to solve the given system of equations by substitution. The system is:

\[
\begin{cases}
-2x + 5y = 7 \\
y = -3x - 2
\end{cases}
\]

### Step 1: Substitute the expression for \( y \) from the second equation into the first equation.

From the second equation, we know that \( y = -3x - 2 \). We substitute this into the first equation:

\[
-2x + 5(-3x - 2) = 7
\]

### Step 2: Simplify and solve for \( x \).

Distribute the 5 on the left-hand side:

\[
-2x + 5(-3x) + 5(-2) = 7
\]
\[
-2x - 15x - 10 = 7
\]
Combine like terms:
\[
-17x - 10 = 7
\]
Add 10 to both sides:
\[
-17x = 17
\]
Solve for \( x \):
\[
x = -1
\]

### Step 3: Substitute \( x = -1 \) back into the second equation to find \( y \).

Now that we know \( x = -1 \), substitute it into the equation \( y = -3x - 2 \):

\[
y = -3(-1) - 2
\]
\[
y = 3 - 2
\]
\[
y = 1
\]

### Solution:
The solution to the system is \( x = -1 \) and \( y = 1 \). Therefore, the system has **one solution**, and the solution is \( (-1, 1) \).

Let me know if you would like further details!

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Here are 5 related questions to expand your understanding:
1. What would happen if both equations were parallel?
2. Can you solve the same system using elimination instead of substitution?
3. What are the conditions for a system to have no solution?
4. How can you verify the solution to this system graphically?
5. How do you solve systems of equations with more than two variables?

**Tip:** Always check your solution by substituting the values back into both original equations to confirm they satisfy both equations.

Solve the system by substitution: 
 -2x + 5y = 7 
 y = -3x - 2.

Learn how to solve the system of equations -2x + 5y = 7 and y = -3x - 2 using substitution. This problem involves algebraic concepts like systems of equations and the substitution method. Ideal for students in Grades 8-10.

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