To determine the equation of a line that passes through the point [tex]\( (2,1) \)[/tex] and is parallel to the line [tex]\( y = 3x - 4 \)[/tex], let’s follow these steps:
1. Identify the slope of the given line:
The slope of the line [tex]\( y = 3x - 4 \)[/tex] is the coefficient of [tex]\( x \)[/tex] which is 3.
2. Use point-slope form of the line equation:
Since the new line must be parallel to [tex]\( y = 3x - 4 \)[/tex], it will have the same slope. The point-slope form of the equation of a line is given by:
[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 in the given point [tex]\( (2,1) \)[/tex] and slope [tex]\( 3 \)[/tex]:
[tex]\[
y - 1 = 3(x - 2)
\][/tex]
3. Simplify the equation:
Distribute the slope on the right side:
[tex]\[
y - 1 = 3x - 6
\][/tex]
Add 1 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 6 + 1
\][/tex]
[tex]\[
y = 3x - 5
\][/tex]
4. Identify the correct option:
The equation of the line that passes through [tex]\( (2, 1) \)[/tex] and is parallel to [tex]\( y = 3x - 4 \)[/tex] is [tex]\( y = 3x - 5 \)[/tex]. This corresponds to option A.
Thus, the answer is:
A. [tex]\( y = 3x - 5 \)[/tex]
