Find an equation for the line that passes through the points \( (-1,4) \) and \( (5,-6) \).

To find the equation of the line that passes through the points \((-1, 4)\) and \( (5, -6)\), we first need to determine the slope (\(m\)) of the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of the points, we get: \[ m = \frac{-6 - 4}{5 - (-1)} = \frac{-10}{6} = -\frac{5}{3} \] Next, we can use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\). Using one of the points, for instance, \((-1, 4)\): \[ y - 4 = -\frac{5}{3}(x + 1) \] Distributing and rearranging gives us: \[ y - 4 = -\frac{5}{3}x - \frac{5}{3}\\ y = -\frac{5}{3}x + \frac{7}{3} \] Thus, the equation of the line is: \[ y = -\frac{5}{3}x + \frac{7}{3} \] Now, here's some fun info: did you know that the concept of the slope of a line was first discussed by mathematicians like Euclid, but it wasn't until the 17th century when René Descartes and Pierre de Fermat laid the groundwork for coordinate geometry? That's why we can now easily calculate slopes using algebra! Also, consider this real-world application: knowing how to find the equation of a line is super useful in various fields, from economics to engineering. For example, in economics, it helps analyze cost and revenue relationships, helping businesses optimize their strategies based on predicted trends. So, next time you calculate a line, remember you’re tapping into a powerful tool used in everyday decision-making!