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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.

Determine the equation of the line through the points (-6, -2) and (7, 7).

This tutorial covers the step-by-step process of finding the equation of a line passing through the points (-6, -2) and (7, 7). Learn how to calculate the slope and apply the point-slope and slope-intercept forms. Suitable for students in Grades 9-11.

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