https://file.aiquickdraw.com/mathbot/file/1726536803576kyyw9nt2.jpg

The system of equations given in the image is:

\[
\begin{cases} 
y = -3x - 3 \\
6x + 2y = -2
\end{cases}
\]

We need to graph the system and determine its solution.

### Step 1: Graph the first equation \( y = -3x - 3 \)
This is already in slope-intercept form \( y = mx + b \), where:
- \( m = -3 \) (the slope),
- \( b = -3 \) (the y-intercept).

- To plot: Start at \( (0, -3) \), then use the slope of \( -3 \) to move down 3 units and right 1 unit.

### Step 2: Rearrange the second equation \( 6x + 2y = -2 \)
Solve for \( y \) to get it in slope-intercept form:

\[
6x + 2y = -2
\]
\[
2y = -6x - 2
\]
\[
y = -3x - 1
\]

This equation also has:
- \( m = -3 \),
- \( b = -1 \) (the y-intercept).

- To plot: Start at \( (0, -1) \), and again use the slope \( -3 \) to move down 3 units and right 1 unit.

### Step 3: Analyze the graphs
Both equations have the same slope \( m = -3 \), meaning the lines are parallel. Since their y-intercepts are different (\( -3 \) and \( -1 \)), these lines will never intersect.

### Conclusion
The system has no solution because the lines are parallel.

Do you need further details or any other clarifications? Here are 5 related questions to explore the topic further:

1. How do you determine the slope and y-intercept from an equation in slope-intercept form?
2. What is the graphical interpretation of parallel lines in a system of equations?
3. How can you tell if a system of equations has infinitely many solutions?
4. What method can be used to solve a system of equations algebraically instead of graphically?
5. How would you handle a system of equations where one or both equations are not in slope-intercept form?

**Tip:** When two lines have the same slope but different y-intercepts, they are always parallel and will never intersect, leading to no solution.

Graph the system of equations and write its solution: y = -3x - 3 and 6x + 2y = -2.

Learn how to solve the system of equations y = -3x - 3 and 6x + 2y = -2 by graphing. This tutorial covers graphing linear equations and understanding when systems have no solutions due to parallel lines. Suitable for students in Grades 8-10.

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