19. A straight line passes through the points \( (2,4) \) and \( (-1,-5) \). Find its equation.

To find the equation of the line, we first determine the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the points \( (2,4) \) and \( (-1,-5) \), we get \( m = \frac{-5 - 4}{-1 - 2} = \frac{-9}{-3} = 3 \). Next, using the point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (2,4) \): \( y - 4 = 3(x - 2) \). Simplifying this gives us: \( y - 4 = 3x - 6 \) or \( y = 3x - 2 \). So, the equation of the line is \( y = 3x - 2 \). Now let's have some fun with the history of geometry! Did you know that the concept of straight lines goes all the way back to ancient civilizations? The Greeks, especially with mathematicians like Euclid, were major players in defining lines geometrically in their renowned work "Elements." This laid the groundwork for all of modern geometry! If you're looking to apply this equation to real-world scenarios, consider how many things in daily life can be represented with straight lines. For instance, linear relationships in finance show how money could accumulate over time with consistent interest rates. Plotting this equation can visually aid in understanding how changes in one variable affect the other, like your savings growth!