Let's analyze the problem step by step.

The problem states that Kenneth and Gary each wrote a linear equation, and Kenneth's equation is represented by the graph, while Gary's equation is based on the table of values.

1. Finding Kenneth's Equation (from the Graph)

From the graph, we can see two points:

• $(0, -2)$ (y-intercept)
• $(1, 1)$

Using these points, we can find the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-2)}{1 - 0} = \frac{3}{1} = 3$

The equation of a line is given by: $y = mx + b$ where $m$ is the slope, and $b$ is the y-intercept. Here, $m = 3$ and $b = -2$. So, Kenneth's equation is: $y = 3x - 2$

2. Finding Gary's Equation (from the Table)

The table provides the following points:

• $(-1, 6)$
• $(0, 4)$
• $(1, 2)$
• $(2, 0)$

To determine if these points form a linear equation, let's check the slope between consecutive points: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 6}{0 - (-1)} = \frac{-2}{1} = -2$

Repeating this for other points:

• Between $(0, 4)$ and $(1, 2)$: $m = \frac{2 - 4}{1 - 0} = \frac{-2}{1} = -2$
• Between $(1, 2)$ and $(2, 0)$: $m = \frac{0 - 2}{2 - 1} = \frac{-2}{1} = -2$

The slope is consistent, so the points form a line. Using the point $(0, 4)$, which is the y-intercept $b = 4$, we have: $y = -2x + 4$

3. The System of Equations

Kenneth's equation: $y = 3x - 2$
Gary's equation: $y = -2x + 4$

4. Matching with Options

We need to convert both equations to a standard form (if needed) to match with the given options: Kenneth's equation can be rewritten as: $3x - y = 2$ Gary's equation can be rewritten as: $2x + y = 4$

Thus, the correct system is: $\boxed{\text{Option C: } \{3x - y = 2, 2x + y = 4\}}$

Would you like more details or have any further questions?

Here are some follow-up questions you might find useful:

1. How can you find the equation of a line if only two points are given?
2. What does the slope of a line represent?
3. How can you determine if two lines are parallel or perpendicular?
4. What is the significance of the y-intercept in the equation of a line?
5. Can you solve the system of equations to find the intersection point?

Tip: To quickly identify the equation of a line from a graph, look for the y-intercept and calculate the slope using any two points on the line.