https://file.aiquickdraw.com/mathbot/file/1727658604903ov70dh5p.jpg

The system of equations provided is:

\[
\begin{aligned}
1. \quad x + 2y &= -4 \\
2. \quad 4x + 8y &= -16
\end{aligned}
\]

We are asked to solve this using substitution and to determine if the system is dependent, inconsistent, or has a unique solution.

### Step 1: Simplify the second equation
Let's first divide the second equation by 4 to see if we can simplify it:
\[
4x + 8y = -16 \quad \Rightarrow \quad x + 2y = -4
\]
This simplifies to the same equation as the first one.

### Step 2: Analyze the system
Since both equations are the same, the system is **dependent**. This means that every solution of the first equation will also be a solution of the second equation, and there are infinitely many solutions.

### Step 3: Express the solution
The solution set can be expressed in terms of one variable. Let's solve for \( x \) in terms of \( y \) using the first equation:
\[
x + 2y = -4 \quad \Rightarrow \quad x = -4 - 2y
\]
Thus, the solution set is:
\[
\boxed{(x, y) = (-4 - 2y, y)}
\]
where \( y \) can be any real number.

### Conclusion:
This is a **dependent system** with infinitely many solutions.

---

Would you like more details on any step? 

Here are 5 related questions for practice:
1. What is the difference between a dependent and an inconsistent system?
2. How can you tell a system of equations is dependent by looking at it?
3. Can you solve a system with substitution if it's inconsistent?
4. How do you graphically represent a dependent system of equations?
5. What does it mean if the solution to a system of equations is a line?

**Tip:** A dependent system means both equations represent the same line, so they overlap infinitely on the graph!

Use the method of substitution to solve the following system of equations. If the system is dependent, express the solution set in terms of one of the variables. Leave all fractional answers in fraction form.

x + 2y = -4
4x + 8y = -16

Learn how to solve a system of linear equations using the substitution method. This problem explores whether the system is dependent or has a unique solution. Includes step-by-step solutions and explanations for high school students in Grades 9-12.

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