Determine the Equation of a Line Parallel to y=3x+4 Through (1/2,1)

To solve this problem, we begin by noting that since the line is parallel to $y = 3x + 4$, it must have the same slope, $m = 3$.

We use the point-slope form of the equation of a line, which is:

$y - y_1 = m(x - x_1)$

Here, the slope $m = 3$ and the line passes through the point $\left(\frac{1}{2}, 1\right)$. Therefore, we substitute these values into the point-slope formula:

$y - 1 = 3\left(x - \frac{1}{2}\right)$

Next, we simplify this equation:

• Distribute the slope $3$ on the right side:
$y - 1 = 3x - \frac{3}{2}$
• Add 1 to both sides to solve for $y$:
$y = 3x - \frac{3}{2} + 1$
• Simplify $-\frac{3}{2} + 1$:
$y = 3x - \frac{1}{2}$

Thus, the equation of the line parallel to $y = 3x + 4$ and passing through the point $\left(\frac{1}{2}, 1\right)$ is:

$y = 3x - \frac{1}{2}$

The corresponding choice is:

$y=3x-\frac{1}{2}$