To find the equation of a line in point-slope form that is perpendicular to [tex]\( y = 3x + 5 \)[/tex] and passes through the point [tex]\( (4, -1) \)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = 3x + 5 \)[/tex]. The coefficient of [tex]\( x \)[/tex] is the slope (m) of this line.
So, the slope of the given line is [tex]\( m = 3 \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line.
Therefore, the slope of the perpendicular line [tex]\( m_{\perp} \)[/tex] is:
[tex]\[
m_{\perp} = -\frac{1}{m} = -\frac{1}{3}
\][/tex]
3. 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]\( (x_1, y_1) \)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope.
Given the point [tex]\( (4, -1) \)[/tex] and the slope [tex]\( -\frac{1}{3} \)[/tex], substitute these values into the point-slope form:
[tex]\[
y - (-1) = -\frac{1}{3}(x - 4)
\][/tex]
4. Simplify the equation:
Simplify the left-hand side by turning [tex]\( y - (-1) \)[/tex] into [tex]\( y + 1 \)[/tex]:
[tex]\[
y + 1 = -\frac{1}{3}(x - 4)
\][/tex]
5. Identify the correct option:
Compare the simplified equation with the given choices:
- A. [tex]\( y + 1 = -\frac{1}{3}(x - 4) \)[/tex]
- B. [tex]\( y - 1 = -3(x + 4) \)[/tex]
- C. [tex]\( y + 1 = 3(x - 4) \)[/tex]
- D. [tex]\( y - 4 = -\frac{1}{3}(x + 1) \)[/tex]
The correct equation in point-slope form is:
A. [tex]\( y + 1 = -\frac{1}{3}(x - 4) \)[/tex]
