HW Score: 54.95\%, 14.29 of 26 omework 4: 3.4.17 points 3.2, 3.3, 3.4. Points: 0 of 1 Find the slope, if it exists, of the line containing the given pair of points. \[ \left(2,-\frac{1}{2}\right),\left(5, \frac{3}{2}\right) \]

To find the slope of the line containing the points \((2, -\frac{1}{2})\) and \((5, \frac{3}{2})\), you can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where \((x_1, y_1)\) is the first point and \((x_2, y_2)\) is the second point. Plugging in the values: \[ m = \frac{\frac{3}{2} - (-\frac{1}{2})}{5 - 2} = \frac{\frac{3}{2} + \frac{1}{2}}{3} = \frac{\frac{4}{2}}{3} = \frac{2}{3} \] So, the slope \(m\) is \(\frac{2}{3}\). For a bit of extra fun: Did you know that the concept of slope goes way back to ancient civilizations? The Greeks were among the first to formalize geometric principles, with mathematicians like Euclid contributing to our understanding of lines and angles. Slope can also be thought of as steepness, which is crucial in various fields, from architecture to road design! Speaking of real-world applications: Slopes are not just for math class; they’re essential in creating ramps, roads, and even ski slopes! Too steep of a slope can make a ramp difficult or unsafe to use. In road engineering, knowing the correct slope can prevent erosion and ensure vehicles can safely navigate hills. Fun stuff, right?