To find the equation of a line that contains the point [tex]\((2, 1)\)[/tex] and is perpendicular to the line [tex]\(y = 3x - 4\)[/tex], we can follow these steps:
1. Find the Slope of the Given Line:
The given line is [tex]\(y = 3x - 4\)[/tex]. The slope of this line is the coefficient of [tex]\(x\)[/tex], which is 3.
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, the slope of the perpendicular line is:
[tex]\[
\text{slope}_{\text{perpendicular}} = -\frac{1}{\text{slope}_{\text{original}}} = -\frac{1}{3}
\][/tex]
3. Use the Point-Slope Form to Find the Equation:
The point-slope form of the equation 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 a point on the line.
Plugging the given point [tex]\((2, 1)\)[/tex] and the perpendicular slope [tex]\(-\frac{1}{3}\)[/tex]:
[tex]\[
y - 1 = -\frac{1}{3}(x - 2)
\][/tex]
4. Solve for the Equation in Slope-Intercept Form:
First, distribute the slope on the right-hand side:
[tex]\[
y - 1 = -\frac{1}{3}x + \frac{2}{3}
\][/tex]
Then, add 1 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{1}{3}x + \frac{2}{3} + 1
\][/tex]
Convert 1 to [tex]\(\frac{3}{3}\)[/tex] to combine the fractions:
[tex]\[
y = -\frac{1}{3}x + \frac{2}{3} + \frac{3}{3}
\][/tex]
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3}
\][/tex]
So, the equation of the line that is perpendicular to [tex]\(y = 3x - 4\)[/tex] and passes through the point [tex]\((2, 1)\)[/tex] is:
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3}
\][/tex]
Hence, the correct option is:
A. [tex]\(y = -\frac{1}{3}x + \frac{5}{3}\)[/tex]
