To find the equation of a line that passes through the point $(8, 2)$ and has a slope of $-\frac{3}{4}$, we can use the point-slope form of the equation of a line:

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

where:

• $m$ is the slope of the line,
• $(x_1, y_1)$ is a point on the line.

Step-by-Step Solution

1. Identify the given values:

   • Slope $m = -\frac{3}{4}$
   • Point $(x_1, y_1) = (8, 2)$

2. Substitute these values into the point-slope formula:

   $y - 2 = -\frac{3}{4}(x - 8)$

3. Simplify the equation (if needed):

   • Distribute the slope on the right side:

     $y - 2 = -\frac{3}{4}x + \frac{3}{4} \cdot 8$

   • Calculate $\frac{3}{4} \cdot 8 = 6$, so:

     $y - 2 = -\frac{3}{4}x + 6$

4. Isolate $y$ to put the equation in slope-intercept form $y = mx + b$:

   $y = -\frac{3}{4}x + 6 + 2$

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

Final Answer

The equation of the line is:

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

Would you like further details, or do you have any questions?

Here are some related questions to deepen your understanding:

1. How would the equation change if the slope were positive?
2. Can you convert the point-slope form to standard form $Ax + By = C$?
3. What if the point was $(0, 5)$ with the same slope? How would the equation look?
4. What does the slope of $-\frac{3}{4}$ indicate about the line's direction?
5. How can we find the x-intercept of the line?

Tip: Always substitute carefully and simplify step-by-step to avoid errors in calculations.