To solve the problem, we need to determine which equation corresponds to the given line on the graph. Here's the step-by-step breakdown:

Step 1: Identify two points on the line

Looking at the graph, two clear points that the line passes through are:

• Point A: (0, 3) (y-intercept)
• Point B: (6, -3)

Step 2: Find the slope of the line

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the points $(0, 3)$ and $(6, -3)$:

$m = \frac{-3 - 3}{6 - 0} = \frac{-6}{6} = -1$

So, the slope of the line is $m = -1$.

Step 3: Write the equation of the line in slope-intercept form

The slope-intercept form of a line is:

$y = mx + b$

We already know the slope $m = -1$ and the y-intercept $b = 3$ (from point A where the line crosses the y-axis). So, the equation is:

$y = -x + 3$

Step 4: Convert the equation to standard form

To convert $y = -x + 3$ to standard form $Ax + By = C$, we rearrange the terms:

$x + y = 3$

Step 5: Multiply both sides by 6 to match the given options

Now, to match the form of the provided options, multiply both sides by 6:

$6x + 6y = 18$

Step 6: Identify the correct answer

The closest option to this equation is the third one:

$3x + 6y = 18$

Therefore, the correct answer is the third option: $\boxed{3x + 6y = 18}$.

Would you like more details, or do you have any questions about this process? Here are some related questions to further your understanding:

1. How do you calculate the slope if the points are not integers?
2. Can you explain why the equation $y = mx + b$ is called the slope-intercept form?
3. How would the graph change if the slope were positive instead of negative?
4. What would happen to the equation if the line passed through the origin?
5. How do you convert an equation from standard form back to slope-intercept form?

Tip: Always start by identifying the slope and y-intercept when trying to derive the equation of a line!