To find the equation of a line that goes through the point [tex]\((2,1)\)[/tex] and is parallel to the line [tex]\( y = 3x - 4 \)[/tex], we need to follow these steps:
1. Determine the Slope:
The slope of the given line [tex]\( y = 3x - 4 \)[/tex] is [tex]\( 3 \)[/tex] (the coefficient of [tex]\( x \)[/tex]).
Since parallel lines have the same slope, the slope of our desired line is also [tex]\( 3 \)[/tex].
2. 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]\( m = 3 \)[/tex].
Plug these values into the point-slope form:
[tex]\[
y - 1 = 3(x - 2)
\][/tex]
3. Simplify the Equation:
Distribute the [tex]\( 3 \)[/tex] on the right-hand side:
[tex]\[
y - 1 = 3x - 6
\][/tex]
Add [tex]\( 1 \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 6 + 1
\][/tex]
[tex]\[
y = 3x - 5
\][/tex]
Therefore, the equation of the line that passes through the point [tex]\((2, 1)\)[/tex] and is parallel to the line [tex]\( y = 3x - 4 \)[/tex] is:
[tex]\[
y = 3x - 5
\][/tex]
So, the correct answer is:
B. [tex]\( y = 3x - 5 \)[/tex]
