To find the equation of the line that passes through the points $(1, 9)$ and $(7, -4)$, we can use the point-slope form of a line:

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

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line.

Step 1: Find the Slope

The slope $m$ is calculated using the formula:

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

Substituting the given points $(x_1, y_1) = (1, 9)$ and $(x_2, y_2) = (7, -4)$:

$m = \frac{-4 - 9}{7 - 1} = \frac{-13}{6}$

Step 2: Use Point-Slope Form

Using the point-slope form with the point $(1, 9)$ and the slope $m = \frac{-13}{6}$:

$y - 9 = \frac{-13}{6}(x - 1)$

Step 3: Simplify to Slope-Intercept Form

Now, let's simplify this equation to the slope-intercept form $y = mx + b$:

$y - 9 = \frac{-13}{6}x + \frac{13}{6}$

Adding 9 to both sides:

$y = \frac{-13}{6}x + \frac{13}{6} + 9$

Convert 9 to a fraction with a denominator of 6:

$y = \frac{-13}{6}x + \frac{13}{6} + \frac{54}{6}$

$y = \frac{-13}{6}x + \frac{67}{6}$

Final Equation

So, the equation of the line in slope-intercept form is:

$y = \frac{-13}{6}x + \frac{67}{6}$

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions:

1. How would you find the equation of a line if given a different pair of points?
2. What is the significance of the slope in the context of a line?
3. How do you convert between point-slope form and slope-intercept form?
4. What does the y-intercept represent in the equation of a line?
5. How can you verify that a point lies on a given line?

Tip: Always double-check your slope calculation, as it's a common source of errors when finding the equation of a line.