To find the equation of a line that is perpendicular to the line [tex]\( y = 3x - 4 \)[/tex] and passes through the point [tex]\((2, 1)\)[/tex], follow these steps:
1. Determine the slope of the given line: The given line is [tex]\( y = 3x -4 \)[/tex]. The slope of this line (denoted [tex]\( m \)[/tex]) is the coefficient of [tex]\( x \)[/tex], which is [tex]\( 3 \)[/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 given line.
- The negative reciprocal of [tex]\( 3 \)[/tex] is [tex]\( -\frac{1}{3} \)[/tex].
3. Use the point-slope form of the equation of a line: The point-slope form is given by
[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.
- Here, [tex]\((x_1, y_1) = (2, 1)\)[/tex] and [tex]\( m = -\frac{1}{3} \)[/tex].
- Substitute these values into the point-slope form:
[tex]\[
y - 1 = -\frac{1}{3}(x - 2)
\][/tex]
4. Simplify the equation: Convert the point-slope form to slope-intercept form ([tex]\( y = mx + b \)[/tex]).
- Distribute the slope [tex]\( -\frac{1}{3} \)[/tex] on the right side:
[tex]\[
y - 1 = -\frac{1}{3}x + \frac{2}{3}
\][/tex]
- Add [tex]\( 1 \)[/tex] to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{3}x + \frac{2}{3} + 1
\][/tex]
- Convert [tex]\( 1 \)[/tex] to a fraction with denominator 3:
[tex]\[
y = -\frac{1}{3}x + \frac{2}{3} + \frac{3}{3}
\][/tex]
- Combine the fractions:
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3}
\][/tex]
Therefore, the equation of the line that passes through the point [tex]\((2, 1)\)[/tex] and is perpendicular to the line [tex]\( y = 3x - 4 \)[/tex] is
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3}.
\][/tex]
Thus, the correct answer is:
A. [tex]\( y = -\frac{1}{3}x + \frac{5}{3} \)[/tex]
