The image shows a graph with the equation of a line passing through several points, including (-5, -5), (-2, -2), (3, 3), and (7, 7). To find the equation of the line, we can use the slope-intercept form of a linear equation:

$y = mx + b$

where:

• $m$ is the slope,
• $b$ is the y-intercept.

Step 1: Find the Slope $m$

The slope $m$ is calculated as:

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

Using the points $(7,7)$ and $(3,3)$:

$m = \frac{7 - 3}{7 - 3} = \frac{4}{4} = 1$

Thus, the slope $m = 1$.

Step 2: Find the y-intercept $b$

Now, substitute the slope $m = 1$ into the equation $y = mx + b$, and use any point on the line to solve for $b$. Let’s use the point $(3,3)$:

$3 = 1(3) + b$ $3 = 3 + b$ $b = 0$

Final Equation

Thus, the equation of the line is:

$y = x$

This means the line has a slope of 1 and passes through the origin (0,0).

Would you like more details on any step, or do you have further questions?

Here are 5 related questions to expand on this:

1. What happens if we choose different points to calculate the slope?
2. How can we verify that the line passes through the origin?
3. What does the slope of a line represent in general?
4. How would the equation change if the y-intercept were not 0?
5. How do we calculate the slope of a vertical or horizontal line?

Tip: Always verify the line’s slope by selecting two distinct points on the graph to ensure consistency.