To determine the equation of a line that is perpendicular to the given line [tex]\(8y - 16 = 5x\)[/tex] and passes through the point [tex]\((5, -5)\)[/tex], follow these steps:
### Step 1: Rewrite the given equation in slope-intercept form
First, rearrange the given equation to the slope-intercept form, [tex]\(y = mx + b\)[/tex]:
[tex]\[
8y - 16 = 5x
\][/tex]
Add 16 to both sides:
[tex]\[
8y = 5x + 16
\][/tex]
Divide every term by 8:
[tex]\[
y = \frac{5}{8}x + 2
\][/tex]
Thus, the slope (m) of the given line is [tex]\(\frac{5}{8}\)[/tex].
### Step 2: Determine the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Hence, the slope of the perpendicular line is:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{\left( \frac{5}{8} \right)} = -\frac{8}{5}
\][/tex]
### Step 3: Use the point-slope form to find the equation of the line
Now, use the point-slope form of the equation [tex]\( y - y_1 = m(x - x_1) \)[/tex] where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((5, -5)\)[/tex] and [tex]\(m\)[/tex] is [tex]\(-\frac{8}{5}\)[/tex]:
[tex]\[
y - (-5) = -\frac{8}{5}(x - 5)
\][/tex]
This simplifies to:
[tex]\[
y + 5 = -\frac{8}{5}(x - 5)
\][/tex]
Distribute the slope term on the right side:
[tex]\[
y + 5 = -\frac{8}{5}x + 8
\][/tex]
Subtract 5 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{8}{5}x + 8 - 5
\][/tex]
[tex]\[
y = -\frac{8}{5}x + 3
\][/tex]
### Conclusion
The equation of the line that is perpendicular to the given line and passes through the point [tex]\((5, -5)\)[/tex] is:
[tex]\[
y = -\frac{8}{5}x + 3
\][/tex]
Thus, the correct answer is [tex]\(\boxed{D}\)[/tex].
