To determine the equation of the line that passes through the points $(9, -4)$ and $(2, -2)$, follow these steps:

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1. Find the slope ($m$) of the line.

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$m = \frac{y_2 - y_1}{x_2 - x_1}.$
Substitute $(x_1, y_1) = (9, -4)$ and $(x_2, y_2) = (2, -2)$:

$m = \frac{-2 - (-4)}{2 - 9}.$

Simplify the numerator and denominator:

$m = \frac{-2 + 4}{2 - 9} = \frac{2}{-7}.$

Thus, the slope is:

$m = -\frac{2}{7}.$

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2. Use the point-slope form of a line.

The point-slope form of a line is:
$y - y_1 = m(x - x_1),$ where $m$ is the slope, and $(x_1, y_1)$ is any point on the line.
We can use one of the given points, say $(9, -4)$, and $m = -\frac{2}{7}$.

Substitute these values into the formula:

$y - (-4) = -\frac{2}{7}(x - 9).$

Simplify the equation:

$y + 4 = -\frac{2}{7}(x - 9).$

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3. Fully simplified point-slope form.

The equation of the line in point-slope form is:

$y + 4 = -\frac{2}{7}(x - 9).$

This is the required equation of the line.

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Let me know if you'd like the equation in slope-intercept form or if you have any questions! Here are 5 related problems:

1. Find the equation of the line that passes through $(3, 5)$ and $(7, 1)$.
2. Write the equation of a vertical line passing through $(4, -2)$.
3. Determine the slope of the line passing through $(1, -6)$ and $(4, 3)$.
4. Convert $y + 4 = -\frac{2}{7}(x - 9)$ to slope-intercept form.
5. Write the equation of a horizontal line passing through $(-3, 5)$.

Tip: Always calculate the slope carefully, as sign errors are common!