Find the equation of the line parallel to $-3x - 2y = 7$ and passing through the point $(\frac{1}{2}, -2)$.

1. Problem Description

Find the equation of the line parallel to $-3x - 2y = 7$ and passing through the point $(\frac{1}{2}, -2)$.

2. Solution Steps

First, rewrite the given equation in slope-intercept form to find its slope.

$-3x - 2y = 7$

$-2y = 3x + 7$

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

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

Since we want a line parallel to this line, the slope of the new line will also be $-\frac{3}{2}$.

We have the slope $m = -\frac{3}{2}$ and a point $(\frac{1}{2}, -2)$. We can use the point-slope form to find the equation of the new line:

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

Where $(x_1, y_1)$ is the given point $(\frac{1}{2}, -2)$. Plugging in the values, we get:

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

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

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

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

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

Now, we can rewrite this in standard form:

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

Multiply by 4 to eliminate fractions:

$4y = -6x - 5$

$6x + 4y = -5$

3. Final Answer

The equation of the line is $6x + 4y = -5$.

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