1. **State the problem:** We are given the line equation $y=\frac{3}{2}x-1$ and need to find the equation of the line perpendicular to it that passes through the point $(3,4)$. 2. **Recall the slope of the given line:** The slope $m$ of the line $y=\frac{3}{2}x-1$ is $\frac{3}{2}$. 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, $$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{3}{2}} = -\frac{2}{3}.$$ 4. **Use point-slope form to find the perpendicular line equation:** The point-slope form is $$y - y_1 = m(x - x_1),$$ where $(x_1,y_1) = (3,4)$ and $m = -\frac{2}{3}$. Substitute values: $$y - 4 = -\frac{2}{3}(x - 3).$$ 5. **Simplify the equation:** $$y - 4 = -\frac{2}{3}x + 2,$$ $$y = -\frac{2}{3}x + 2 + 4,$$ $$y = -\frac{2}{3}x + 6.$$ **Final answer:** The equation of the line perpendicular to $y=\frac{3}{2}x-1$ passing through $(3,4)$ is $$y = -\frac{2}{3}x + 6.$$