1. **State the problem:** Find the equation of the line $\ell$ in the $xy$-plane that passes through the point $(3,4)$ and makes an angle of $70^\circ$ with the positive $x$-axis. 2. **Formula and explanation:** The slope $m$ of a line making an angle $\theta$ with the positive $x$-axis is given by: $$m = \tan \theta$$ The point-slope form of the line equation is: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1)$ is a point on the line. 3. **Calculate the slope:** $$m = \tan 70^\circ \approx 2.747$$ 4. **Write the equation using the point $(3,4)$:** $$y - 4 = 2.747(x - 3)$$ 5. **Simplify the equation:** $$y - 4 = 2.747x - 8.241$$ $$y = 2.747x - 8.241 + 4$$ $$y = 2.747x - 4.241$$ **Final answer:** $$y = 2.747x - 4.241$$