Question: Determine a equation for a line through $(-5, 2)$ and parallel to $2x - 3y = 6$
1. **State the problem:** Find the equation of a line passing through the point $(-5, 2)$ and parallel to the line given by $2x - 3y = 6$.
2. **Recall the formula and rules:**
- Lines that are parallel have the same slope.
- First, find the slope of the given line by rewriting it in slope-intercept form $y = mx + b$.
3. **Find the slope of the given line:**
$$2x - 3y = 6$$
Rearrange to solve for $y$:
$$-3y = -2x + 6$$
$$y = \frac{-2x + 6}{-3} = \frac{2}{3}x - 2$$
So, the slope $m$ of the given line is $\frac{2}{3}$.
4. **Use the point-slope form for the new line:**
The new line has slope $m = \frac{2}{3}$ and passes through $(-5, 2)$.
The point-slope form is:
$$y - y_1 = m(x - x_1)$$
Substitute $x_1 = -5$, $y_1 = 2$, and $m = \frac{2}{3}$:
$$y - 2 = \frac{2}{3}(x + 5)$$
5. **Simplify the equation:**
$$y - 2 = \frac{2}{3}x + \frac{10}{3}$$
$$y = \frac{2}{3}x + \frac{10}{3} + 2$$
Convert $2$ to thirds:
$$2 = \frac{6}{3}$$
So,
$$y = \frac{2}{3}x + \frac{10}{3} + \frac{6}{3} = \frac{2}{3}x + \frac{16}{3}$$
6. **Match with the options:**
Option b is:
$$y = \frac{2}{3}x + \frac{16}{3}$$
which matches our result.
**Final answer:** Option b. $y = \frac{2}{3}x + \frac{16}{3}$