Sure, let's find the equation of the line step-by-step.
1. Identify the Slope of the Given Line:
The given line is [tex]\(y + 5 = \frac{7}{5}(x - 16)\)[/tex].
This is in the point-slope form [tex]\((y - y_1 = m(x - x_1))\)[/tex], where [tex]\(m\)[/tex] is the slope of the line.
Here, the slope [tex]\(m_1\)[/tex] is [tex]\(\frac{7}{5}\)[/tex].
2. Find the Slope of the Perpendicular Line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
Therefore, if the original slope [tex]\(m_1 = \frac{7}{5}\)[/tex], then the slope [tex]\(m_2\)[/tex] of the perpendicular line is:
[tex]\[
m_2 = -\frac{1}{\frac{7}{5}} = -\frac{5}{7}
\][/tex]
3. Using the Point-Slope Form to Find the Perpendicular Line:
The point we need the line to pass through is [tex]\((14, 0)\)[/tex].
The point-slope form of a line is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point.
Plugging in the values:
[tex]\[
y - 0 = -\frac{5}{7}(x - 14)
\][/tex]
4. Simplify the Equation:
Distributing the slope [tex]\(-\frac{5}{7}\)[/tex] on the right-hand side:
[tex]\[
y = -\frac{5}{7}x + -\frac{5}{7} \cdot -14
\][/tex]
Calculate [tex]\(\frac{5}{7} \cdot 14\)[/tex]:
[tex]\[
-\frac{5}{7} \cdot 14 = -10
\][/tex]
So the equation becomes:
[tex]\[
y = -\frac{5}{7}x + 10
\][/tex]
Therefore, the equation of the line perpendicular to the given line and passing through the point [tex]\((14, 0)\)[/tex] is:
[tex]\[
y = -\frac{5}{7}x + 10
\][/tex]
The provided options are not correct, as none of them match [tex]\(y = -\frac{5}{7}x + 10\)[/tex]. Therefore, the answer is:
[tex]\[
\text{None of these are correct.}
\][/tex]
