https://file.aiquickdraw.com/mathbot/file/1726688886083gk2wmc5e.jpg

The system of equations provided is:

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

Let's solve and graph the system step by step.

### Step 1: Graph the first equation \( y = 3x - 3 \)
This is already in slope-intercept form \( y = mx + b \), where \( m = 3 \) (the slope) and \( b = -3 \) (the y-intercept). The line passes through the point \( (0, -3) \) and has a slope of 3, meaning for every 1 unit to the right, the line goes up 3 units.

### Step 2: Manipulate the second equation
We rewrite the second equation \( 6x - 2y = 6 \) in slope-intercept form \( y = mx + b \).

\[
6x - 2y = 6 \quad \text{(subtract \( 6x \) from both sides)}
\]
\[
-2y = -6x + 6 \quad \text{(divide the whole equation by -2)}
\]
\[
y = 3x - 3
\]

This gives the same equation as the first one: \( y = 3x - 3 \).

### Step 3: Determine the solution
Since both equations represent the same line, the system has **infinitely many solutions**. Any point on the line \( y = 3x - 3 \) satisfies both equations.

### Conclusion
The solution to this system is **infinitely many solutions**, meaning every point on the line \( y = 3x - 3 \) is a solution.

Would you like more details, or do you have any questions about this?

Here are 5 related questions to further explore this topic:
1. How do you interpret the slope and y-intercept in real-world problems?
2. How can you determine if two lines are parallel, intersect, or coincident without graphing them?
3. What is the difference between consistent, inconsistent, and dependent systems of equations?
4. How can you solve a system of equations algebraically, such as by substitution or elimination?
5. What are some applications of systems of equations in science or engineering?

**Tip:** When two equations in a system simplify to the same equation, this indicates that the system has infinitely many solutions.

Graph the system below and write its solution.
 y = 3x - 3 
 6x - 2y = 6

Learn how to solve and graph a system of linear equations: y = 3x - 3 and 6x - 2y = 6. Understand graphing techniques and determine if the system has one solution, no solution, or infinitely many solutions.

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