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 given equation is [tex]\(y = 3x - 4\)[/tex]. This line is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. From this equation, we see that the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex].
2. Understand that parallel lines have the same slope:
Since parallel lines have the same slope, the line we are looking for will also have a slope of [tex]\(3\)[/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 the given point [tex]\((2,1)\)[/tex] and [tex]\(m\)[/tex] is the slope. Substituting the known values, we get:
[tex]\[
y - 1 = 3(x - 2)
\][/tex]
4. Simplify the equation to get it into slope-intercept form ([tex]\(y = mx + b\)[/tex]):
Expand the right side:
[tex]\[
y - 1 = 3x - 6
\][/tex]
Isolate [tex]\(y\)[/tex] by adding 1 to both sides:
[tex]\[
y = 3x - 6 + 1
\][/tex]
Simplify:
[tex]\[
y = 3x - 5
\][/tex]
So, the equation of the line that contains the point [tex]\((2,1)\)[/tex] and is parallel to the line [tex]\(y = 3x - 4\)[/tex] is:
[tex]\[
y = 3x - 5
\][/tex]
The correct option is:
A. [tex]\(y = 3x - 5\)[/tex]
