To find the equation of the line that passes through the points $(-12, 8)$ and $(9, 1)$, we use the slope-intercept form:

$y = mx + b$

Step 1: Find the slope ($m$):

The formula for the slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points $(x_1, y_1) = (-12, 8)$ and $(x_2, y_2) = (9, 1)$: $m = \frac{1 - 8}{9 - (-12)} = \frac{-7}{9 + 12} = \frac{-7}{21} = -\frac{1}{3}$

Step 2: Use the slope and a point to find the y-intercept ($b$):

The slope-intercept form is $y = mx + b$. Using the point $(-12, 8)$: $8 = -\frac{1}{3}(-12) + b$

Simplify: $8 = \frac{12}{3} + b$ $8 = 4 + b$ $b = 8 - 4 = 4$

Step 3: Write the equation of the line:

Substitute $m = -\frac{1}{3}$ and $b = 4$ into $y = mx + b$: $y = -\frac{1}{3}x + 4$

Final Answer:

The equation of the line is: $y = -\frac{1}{3}x + 4$

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Would you like a graph of this line or further clarification?

Here are 5 related questions for practice:

1. How do you find the slope of a line given two points?
2. What is the importance of the y-intercept in the slope-intercept form?
3. Can you convert this equation into standard form ($Ax + By = C$)?
4. How would the equation change if the line were parallel to this one but passed through a different point?
5. How do you find the equation of a line perpendicular to this one passing through the same points?

Tip: Always simplify fractions when calculating the slope for cleaner results.