1. **State the problem:** We are given the line $y=\frac{2}{3}x - 1$ and a point $(3,4)$. We need to find the equation of the line perpendicular to the given line that passes through the point $(3,4)$. 2. **Recall the slope of the given line:** The slope $m$ of the line $y=\frac{2}{3}x - 1$ is $\frac{2}{3}$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, $$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2}.$$ 4. **Use point-slope form:** The equation of a line with slope $m_{\perp}$ passing through $(x_1,y_1) = (3,4)$ is $$y - y_1 = m_{\perp}(x - x_1).$$ Substitute values: $$y - 4 = -\frac{3}{2}(x - 3).$$ 5. **Simplify the equation:** $$y - 4 = -\frac{3}{2}x + \frac{9}{2}.$$ Add 4 to both sides: $$y = -\frac{3}{2}x + \frac{9}{2} + 4.$$ Convert 4 to fraction with denominator 2: $$4 = \frac{8}{2}.$$ So, $$y = -\frac{3}{2}x + \frac{9}{2} + \frac{8}{2} = -\frac{3}{2}x + \frac{17}{2}.$$ **Final answer:** $$\boxed{y = -\frac{3}{2}x + \frac{17}{2}}.$$