To find the equation of the line passing through the points $(8, -3)$ and $(4, 2)$, we need to follow these steps:

Step 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 coordinates $(8, -3)$ and $(4, 2)$ into the formula:

$m = \frac{2 - (-3)}{4 - 8} = \frac{2 + 3}{-4} = \frac{5}{-4} = -\frac{5}{4}$

So, the slope $m$ is $-\frac{5}{4}$.

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

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

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

We can use either of the points. Let's use $(8, -3)$. Substituting the slope $m = -\frac{5}{4}$ and the point $(x_1, y_1) = (8, -3)$, we get:

$y - (-3) = -\frac{5}{4}(x - 8)$

Simplify the equation:

$y + 3 = -\frac{5}{4}(x - 8)$

Distribute the slope on the right-hand side:

$y + 3 = -\frac{5}{4}x + 10$

Step 3: Solve for $y$

Now, subtract 3 from both sides to isolate $y$:

$y = -\frac{5}{4}x + 10 - 3$

$y = -\frac{5}{4}x + 7$

Final Answer:

The equation of the line passing through the points $(8, -3)$ and $(4, 2)$ is:

$y = -\frac{5}{4}x + 7$

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Would you like more details about any of the steps? Here are some related questions to explore further:

1. How do you convert the point-slope form to slope-intercept form?
2. How can you calculate the distance between two points on the line?
3. What is the significance of the slope in real-world problems?
4. Can you find the equation of a line given only the slope and one point?
5. How would the equation change if the points were closer or farther apart?

Tip: Always check the slope calculation first to ensure you have the correct steepness for the line before writing the equation!