To determine the equation of a line that passes through the point [tex]\((8, -5)\)[/tex] and is parallel to a given line, we need to follow a few steps. Given that parallel lines have the same slope, let's denote the slope of the given line as [tex]\(m\)[/tex].
Step 1: Identify the slope.
Here, we know that the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Step 2: Use the point-slope form of the line equation.
The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
We have:
- Slope [tex]\(m = \frac{3}{4}\)[/tex]
- Point [tex]\((x_1, y_1) = (8, -5)\)[/tex]
Substitute these values into the point-slope form:
[tex]\[ y - (-5) = \frac{3}{4}(x - 8) \][/tex]
[tex]\[ y + 5 = \frac{3}{4}(x - 8) \][/tex]
Step 3: Simplify to find the y-intercept.
First, distribute the slope on the right side:
[tex]\[ y + 5 = \frac{3}{4}x - \frac{3}{4} \cdot 8 \][/tex]
[tex]\[ y + 5 = \frac{3}{4}x - 6 \][/tex]
Now, isolate [tex]\(y\)[/tex] by subtracting 5 from both sides:
[tex]\[ y = \frac{3}{4}x - 6 - 5 \][/tex]
[tex]\[ y = \frac{3}{4}x - 11 \][/tex]
Thus, the equation of the line is:
[tex]\[ y = \frac{3}{4}x - 11 \][/tex]
Step 4: Match this with the provided options.
The correct option is:
[tex]\[ \boxed{D. \quad y = \frac{3}{4} x - 11} \][/tex]
Therefore, the equation of the line passing through the point [tex]\((8, -5)\)[/tex] and parallel to the given line is [tex]\(\boxed{y = \frac{3}{4} x - 11}\)[/tex].
