1. **State the problem:** Find the equation of the line passing through the point $(-5, 12)$ and perpendicular to the line given by $$y = \frac{2}{3}x + 14.$$ 2. **Recall the formula and rules:** The slope of the given line is $m = \frac{2}{3}$. The slope of a line perpendicular to it is the negative reciprocal, so $$m_{\perp} = -\frac{3}{2}.$$ 3. **Use point-slope form:** The equation of a line with slope $m$ passing through $(x_1, y_1)$ is $$y - y_1 = m(x - x_1).$$ Substitute $m = -\frac{3}{2}$ and point $(-5, 12)$: $$y - 12 = -\frac{3}{2}(x + 5).$$ 4. **Simplify:** $$y - 12 = -\frac{3}{2}x - \frac{15}{2}.$$ Add 12 to both sides: $$y = -\frac{3}{2}x - \frac{15}{2} + 12.$$ Convert 12 to halves: $$12 = \frac{24}{2},$$ so $$y = -\frac{3}{2}x + \frac{9}{2}.$$ 5. **Convert to standard form:** Multiply both sides by 2 to clear denominators: $$2y = -3x + 9.$$ Rewrite: $$3x + 2y = 9.$$ 6. **Check options:** The equation $3x + 2y = 9$ matches the first option. **Final answer:** $$3x + 2y = 9.$$