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We are tasked with solving a system of equations using the substitution method. The system provided is:

\[
\begin{cases} 
x + 5y = 22 \\
y = -4x - 7 
\end{cases}
\]

### Step 1: Substitute the expression for \( y \) into the first equation.
From the second equation, we already have \( y = -4x - 7 \). We substitute this into the first equation:

\[
x + 5(-4x - 7) = 22
\]

### Step 2: Simplify the equation.
Distribute the 5:

\[
x + 5(-4x) + 5(-7) = 22
\]
\[
x - 20x - 35 = 22
\]
\[
-19x - 35 = 22
\]

### Step 3: Solve for \( x \).
Add 35 to both sides:

\[
-19x = 22 + 35
\]
\[
-19x = 57
\]

Now, divide both sides by -19:

\[
x = \frac{57}{-19}
\]
\[
x = -3
\]

### Step 4: Solve for \( y \).
Now substitute \( x = -3 \) into the second equation \( y = -4x - 7 \):

\[
y = -4(-3) - 7
\]
\[
y = 12 - 7
\]
\[
y = 5
\]

### Final Answer:
The solution to the system of equations is \( (x, y) = (-3, 5) \).

Would you like any further details or clarifications?

### Related Questions:
1. What is the general approach to solving systems of equations using the substitution method?
2. How do you handle systems of equations when both are nonlinear?
3. How does the substitution method compare to the elimination method for solving systems of equations?
4. What happens if a system of equations has no solution or infinite solutions?
5. How do you verify your solution for a system of equations?

### Tip:
Always double-check your substitution step, as small sign errors can lead to incorrect final answers.

Solve the following system of equations with the substitution method: {x + 5y = 22, y = -4x - 7}.

Learn how to solve a system of linear equations using the substitution method. This step-by-step solution covers the equations x + 5y = 22 and y = -4x - 7, focusing on algebraic manipulation. Suitable for students in Grades 8-10.

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