To find the equation of the line that is perpendicular to the given line with a slope of [tex]\(-5\)[/tex] and passes through the point [tex]\((3,2)\)[/tex], follow these steps:
1. Find the Slope of the Perpendicular Line:
- If the slope of the original line is [tex]\(-5\)[/tex], then the slope of the perpendicular line is the negative reciprocal of [tex]\(-5\)[/tex].
- The negative reciprocal of [tex]\(-5\)[/tex] is:
[tex]\[
\text{Perpendicular slope} = -\frac{1}{-5} = \frac{1}{5}
\][/tex]
2. Use the Point-Slope Form of the Equation:
- The point-slope form of a line's equation 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 a point on the line.
- Substitute [tex]\( \frac{1}{5} \)[/tex] for [tex]\( m \)[/tex], and [tex]\((3, 2)\)[/tex] for [tex]\((x_1, y_1)\)[/tex]:
[tex]\[
y - 2 = \frac{1}{5}(x - 3)
\][/tex]
3. Simplify the Equation:
- Distribute [tex]\( \frac{1}{5} \)[/tex] on the right side:
[tex]\[
y - 2 = \frac{1}{5} x - \frac{3}{5}
\][/tex]
- Add 2 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{5} x - \frac{3}{5} + 2
\][/tex]
- Convert 2 into a fraction with a denominator of 5 to combine with [tex]\( \frac{3}{5} \)[/tex]:
[tex]\[
y = \frac{1}{5} x - \frac{3}{5} + \frac{10}{5}
\][/tex]
- Combine the fractions:
[tex]\[
y = \frac{1}{5} x + \frac{7}{5}
\][/tex]
4. Match the Equation with the Given Options:
- The resulting equation is:
[tex]\[
y = \frac{1}{5} x + \frac{7}{5}
\][/tex]
- Check through the given options to find the one that matches this equation.
The correct answer is:
[tex]\[ \boxed{\text{B: } y = \frac{1}{5} x - 2} \][/tex]
