To determine the equation of a line that contains the point [tex]\((2,1)\)[/tex] and is parallel to the line [tex]\(y = 3x - 4\)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(y = 3x - 4\)[/tex]. This line is in slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Therefore, the slope ([tex]\(m\)[/tex]) of this line is 3.
2. Parallel lines have the same slope:
Since the new line is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also 3.
3. Use the point-slope form of a line equation:
The point-slope form is [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.
We are given the point [tex]\((2, 1)\)[/tex] and the slope is 3. Plugging in these values, we get:
[tex]\[
y - 1 = 3(x - 2)
\][/tex]
4. Simplify the equation:
Distribute the slope (3) on the right-hand side:
[tex]\[
y - 1 = 3x - 6
\][/tex]
Add 1 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = 3x - 6 + 1
\][/tex]
[tex]\[
y = 3x - 5
\][/tex]
5. Compare the result with the given options:
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].
Therefore, the correct answer is:
B. [tex]\(y=3x-5\)[/tex]
