The problem asks to find the equation of a line that is parallel to the line $-3x - 2y = 7$ and passes through the point $(\frac{1}{2}, -2)$.

1. Problem Description

The problem asks to find the equation of a line that is parallel to the line $-3x - 2y = 7$ and passes through the point $(\frac{1}{2}, -2)$.

2. Solution Steps

First, we need to find the slope of the given line. We can rewrite the equation in slope-intercept form, $y = mx + b$, where $m$ is the slope.

$-3x - 2y = 7$

$-2y = 3x + 7$

$y = -\frac{3}{2}x - \frac{7}{2}$

The slope of the given line is $-\frac{3}{2}$.

Since the line we are looking for is parallel to the given line, it has the same slope. Therefore, the slope of the new line is also $-\frac{3}{2}$.

Now we have the slope $m = -\frac{3}{2}$ and a point $(\frac{1}{2}, -2)$ that the line passes through. We can use the point-slope form of a line:

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

where $(x_1, y_1)$ is the given point.

Substituting the values, we have:

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

$y + 2 = -\frac{3}{2}x + \frac{3}{4}$

$y = -\frac{3}{2}x + \frac{3}{4} - 2$

$y = -\frac{3}{2}x + \frac{3}{4} - \frac{8}{4}$

$y = -\frac{3}{2}x - \frac{5}{4}$

To write the equation in standard form, we can multiply by 4 to eliminate fractions:

$4y = -6x - 5$

$6x + 4y = -5$

3. Final Answer

The equation of the line is $6x + 4y = -5$ or $y = -\frac{3}{2}x - \frac{5}{4}$.