1. **State the problem:** We are given the line equation $$y=\frac{2}{3}x - 1$$ 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 $$m_1$$ of the given line is the coefficient of $$x$$, which is $$\frac{2}{3}$$. 3. **Find the slope of the perpendicular line:** The slope $$m_2$$ of a line perpendicular to another with slope $$m_1$$ is the negative reciprocal of $$m_1$$. So, $$m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2}$$. 4. **Use point-slope form:** The equation of a line with slope $$m$$ passing through point $$(x_1,y_1)$$ is $$y - y_1 = m(x - x_1)$$. Substitute $$m = -\frac{3}{2}$$ and point $$(3,4)$$: $$y - 4 = -\frac{3}{2}(x - 3)$$. 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 $$\frac{8}{2}$$: $$y = -\frac{3}{2}x + \frac{9}{2} + \frac{8}{2} = -\frac{3}{2}x + \frac{17}{2}$$. **Final answer:** $$y = -\frac{3}{2}x + \frac{17}{2}$$