The problem in the image asks to determine the equation of a line passing through the points (-6, -2) and (7, 7).

We can find the equation of the line using the point-slope form of the equation:

$y - y_1 = m(x - x_1)$

Step 1: Find the slope $m$

The slope of the 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 given points (-6, -2) and (7, 7):

$m = \frac{7 - (-2)}{7 - (-6)} = \frac{7 + 2}{7 + 6} = \frac{9}{13}$

Step 2: Use the point-slope form

Using the point-slope form and substituting one of the points, say (-6, -2), and the slope $m = \frac{9}{13}$:

$y - (-2) = \frac{9}{13}(x - (-6))$

Simplify:

$y + 2 = \frac{9}{13}(x + 6)$

Step 3: Convert to slope-intercept form $y = mx + b$

Distribute the slope on the right-hand side:

$y + 2 = \frac{9}{13}x + \frac{9}{13} \times 6 = \frac{9}{13}x + \frac{54}{13}$

Subtract 2 from both sides to isolate $y$:

$y = \frac{9}{13}x + \frac{54}{13} - 2$

Convert 2 to a fraction with denominator 13:

$y = \frac{9}{13}x + \frac{54}{13} - \frac{26}{13}$

Simplify:

$y = \frac{9}{13}x + \frac{28}{13}$

Thus, the equation of the line is:

$y = \frac{9}{13}x + \frac{28}{13}$

Would you like further details or clarification on this? Here are some related questions to practice:

1. What is the slope of a line through the points (3, 5) and (-1, 2)?
2. How do you derive the equation of a line in standard form?
3. How does changing the slope affect the steepness of a line?
4. What is the equation of a line parallel to this one, passing through a different point?
5. How can we check if a third point lies on this line?

Tip: Always check your final equation by plugging in both points to see if they satisfy the equation.