To find the equation of the line that is perpendicular to the line [tex]\(y = \frac{3}{5}x + 10\)[/tex] and passes through the point [tex]\((15, -5)\)[/tex], we need to follow these steps:
1. Determine the slope of the given line:
The given line is [tex]\(y = \frac{3}{5}x + 10\)[/tex]. The coefficient of [tex]\(x\)[/tex] is the slope of the line. So, the slope [tex]\(m\)[/tex] of the given line is:
[tex]\[
m = \frac{3}{5}
\][/tex]
2. Find the slope of the perpendicular line:
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. If [tex]\(m_1\)[/tex] is the slope of the first line, then the slope [tex]\(m_2\)[/tex] of the line perpendicular to it can be found using:
[tex]\[
m_2 = -\frac{1}{m_1}
\][/tex]
Here, [tex]\(m_1 = \frac{3}{5}\)[/tex], so:
[tex]\[
m_2 = -\frac{1}{\frac{3}{5}} = -\frac{5}{3}
\][/tex]
So, the slope of the perpendicular line is [tex]\(-\frac{5}{3}\)[/tex].
3. Use the point-slope form of the line equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We have:
[tex]\[
(x_1, y_1) = (15, -5)
\][/tex]
and the perpendicular slope [tex]\(m = -\frac{5}{3}\)[/tex]. Plugging these values in, we get:
[tex]\[
y - (-5) = -\frac{5}{3}(x - 15)
\][/tex]
Simplifying, this becomes:
[tex]\[
y + 5 = -\frac{5}{3}(x - 15)
\][/tex]
4. Simplify the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
Distribute [tex]\(-\frac{5}{3}\)[/tex] on the right-hand side:
[tex]\[
y + 5 = -\frac{5}{3}x + 25
\][/tex]
Subtract 5 from both sides:
[tex]\[
y = -\frac{5}{3}x + 20
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\(y = \frac{3}{5}x + 10\)[/tex] and passes through the point [tex]\((15, -5)\)[/tex] is:
[tex]\[
y = -\frac{5}{3}x + 20
\][/tex]
From the given options, the correct answer is:
[tex]\[
y = -\frac{5}{3}x + 20
\][/tex]
