To determine the equation of a line that passes through the point [tex]\(\left(\frac{1}{3}, 7\right)\)[/tex] and is parallel to the line [tex]\(y = 3x - 4\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\(y = 3x - 4\)[/tex]. This is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Here, the slope [tex]\(m\)[/tex] is 3.
2. Use the point-slope form of the equation:
The point-slope form of a line's equation 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. For our point [tex]\(\left(\frac{1}{3}, 7\right)\)[/tex] and slope [tex]\(m = 3\)[/tex], substitute these values into the point-slope form:
[tex]\[
y - 7 = 3\left(x - \frac{1}{3}\right)
\][/tex]
3. Simplify the equation:
To convert this into the slope-intercept form ([tex]\(y = mx + b\)[/tex]), distribute and simplify:
[tex]\[
y - 7 = 3x - 3 \cdot \frac{1}{3}
\][/tex]
[tex]\[
y - 7 = 3x - 1
\][/tex]
Add 7 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = 3x - 1 + 7
\][/tex]
[tex]\[
y = 3x + 6
\][/tex]
4. Resulting equation:
The equation of the line parallel to [tex]\(y = 3x - 4\)[/tex] and passing through the point [tex]\(\left(\frac{1}{3}, 7\right)\)[/tex] is:
[tex]\[
y = 3x + 6
\][/tex]
5. Conclusion:
Comparing this with the given options:
- [tex]\(y = 3x + 7\)[/tex]
- [tex]\(y = -3x - 7\)[/tex]
- [tex]\(y = 3x + 6\)[/tex]
- [tex]\(y = -3x - 6\)[/tex]
The correct equation is [tex]\(y = 3x + 6\)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{y = 3x + 6}
\][/tex]
