https://file.aiquickdraw.com/mathbot/file/1728935064227ycijx9ig.jpg

The image shows two systems of linear equations, labeled as options C and D:

**C:**
\[
\begin{aligned}
3x - y &= 2 \\
x + 2y &= 4
\end{aligned}
\]

**D:**
\[
\begin{aligned}
3x + y &= -2 \\
x + 2y &= 4
\end{aligned}
\]

To solve each system of equations, we can use either substitution or elimination methods. I'll solve both systems and compare the results.

### Option C:
The system is:
\[
\begin{aligned}
3x - y &= 2 \quad (1) \\
x + 2y &= 4 \quad (2)
\end{aligned}
\]

1. Solve equation (1) for \( y \):
\[
y = 3x - 2
\]

2. Substitute \( y = 3x - 2 \) into equation (2):
\[
x + 2(3x - 2) = 4
\]
\[
x + 6x - 4 = 4
\]
\[
7x - 4 = 4
\]
\[
7x = 8
\]
\[
x = \frac{8}{7}
\]

3. Substitute \( x = \frac{8}{7} \) into \( y = 3x - 2 \):
\[
y = 3\left(\frac{8}{7}\right) - 2 = \frac{24}{7} - 2 = \frac{24}{7} - \frac{14}{7} = \frac{10}{7}
\]

So, the solution for option C is:
\[
x = \frac{8}{7}, \quad y = \frac{10}{7}
\]

### Option D:
The system is:
\[
\begin{aligned}
3x + y &= -2 \quad (1) \\
x + 2y &= 4 \quad (2)
\end{aligned}
\]

1. Solve equation (1) for \( y \):
\[
y = -3x - 2
\]

2. Substitute \( y = -3x - 2 \) into equation (2):
\[
x + 2(-3x - 2) = 4
\]
\[
x - 6x - 4 = 4
\]
\[
-5x - 4 = 4
\]
\[
-5x = 8
\]
\[
x = -\frac{8}{5}
\]

3. Substitute \( x = -\frac{8}{5} \) into \( y = -3x - 2 \):
\[
y = -3\left(-\frac{8}{5}\right) - 2 = \frac{24}{5} - 2 = \frac{24}{5} - \frac{10}{5} = \frac{14}{5}
\]

So, the solution for option D is:
\[
x = -\frac{8}{5}, \quad y = \frac{14}{5}
\]

### Summary:
- **Option C** gives the solution \( x = \frac{8}{7}, y = \frac{10}{7} \)
- **Option D** gives the solution \( x = -\frac{8}{5}, y = \frac{14}{5} \)

Let me know if you'd like further explanation or have any other questions! Here are five related questions to expand on the topic:

1. What is the difference between the substitution and elimination methods in solving systems of equations?
2. How can you verify the solutions of a system of equations?
3. Are there any graphical methods to solve systems of linear equations?
4. How would you solve a system of equations with three variables instead of two?
5. What does it mean if a system of equations has no solution or infinite solutions?

**Tip:** When solving systems of equations, always double-check your substitution or elimination steps to avoid small algebraic errors that can lead to incorrect answers!

Solve the system of linear equations: 3x - y = 2 and x + 2y = 4.

This tutorial explains how to solve a system of linear equations: 3x - y = 2 and x + 2y = 4 using substitution and elimination methods. The example walks through each step of solving for both x and y. Suitable for students in Grades 8-10.

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