To find the equation of the line that passes through the points $(-4, 7)$ and $(2, -1)$, we can use the point-slope formula and calculate the slope first:

Step 1: Calculate the slope ($m$)

The slope formula is:

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

Substituting the given points: $m = \frac{-1 - 7}{2 - (-4)} = \frac{-8}{6} = -\frac{4}{3}$

Step 2: Use the point-slope form of the equation

The point-slope formula is:

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

We can use either of the points, say $(-4, 7)$: $y - 7 = -\frac{4}{3}(x - (-4))$

Simplify: $y - 7 = -\frac{4}{3}(x + 4)$

Step 3: Simplify to slope-intercept form ($y = mx + b$)

Distribute the slope: $y - 7 = -\frac{4}{3}x - \frac{16}{3}$

Add 7 to both sides (convert $7$ to $\frac{21}{3}$): $y = -\frac{4}{3}x - \frac{16}{3} + \frac{21}{3}$

Simplify: $y = -\frac{4}{3}x + \frac{5}{3}$

Final Answer:

The equation of the line is: $y = -\frac{4}{3}x + \frac{5}{3}$

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Do you want a breakdown of how to check this solution? Here are 5 related questions to consider:

1. How do you verify if both points satisfy the equation of the line?
2. Can you rewrite the equation in standard form $Ax + By = C$?
3. What is the importance of slope in identifying the steepness of a line?
4. How does the midpoint between $(-4, 7)$ and $(2, -1)$ relate to this line?
5. How would the equation change if the slope were positive instead?

Tip: Always double-check slope calculations carefully to avoid sign errors when subtracting.