To find the equation of the line passing through the points $(-2, 4)$ and $(3, -1)$, we will use the point-slope form of the equation of a line:

1. Find the slope ($m$):

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 the given points $(-2, 4)$ and $(3, -1)$: $m = \frac{-1 - 4}{3 - (-2)} = \frac{-1 - 4}{3 + 2} = \frac{-5}{5} = -1$

So, the slope is $m = -1$.

2. Use the point-slope form:

The point-slope form of a line is: $y - y_1 = m(x - x_1)$

Substitute $m = -1$ and one of the points, say $(-2, 4)$: $y - 4 = -1(x - (-2))$ $y - 4 = -1(x + 2)$

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

Distribute and simplify: $y - 4 = -x - 2$ $y = -x + 2$

Final Equation:

The equation of the line is: $y = -x + 2$

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Would you like a detailed explanation of how to verify the equation? Here are some related questions:

1. How do you verify that both points satisfy the equation $y = -x + 2$?
2. Can you find the equation of the line if the slope was different, say $m = 2$?
3. How do you convert $y = -x + 2$ into standard form $Ax + By = C$?
4. What is the significance of the slope and y-intercept of the equation $y = -x + 2$?
5. Can you graph the equation $y = -x + 2$ on a coordinate plane?

Tip: Always use the slope formula carefully to ensure the correct signs for subtraction.