Find an equation for the line that passes through the points \( (1,-6) \) and \( (-5,2) \).

To find the equation of the line passing through the points \( (1,-6) \) and \( (-5,2) \), we first calculate the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-6)}{-5 - 1} = \frac{8}{-6} = -\frac{4}{3} \] Now using the point-slope form of the equation \( y - y_1 = m(x - x_1) \), we can choose one of the points, say \( (1, -6) \): \[ y - (-6) = -\frac{4}{3}(x - 1) \] Simplifying this gives: \[ y + 6 = -\frac{4}{3}x + \frac{4}{3} \] Then, rearranging to get the standard form: \[ y = -\frac{4}{3}x + \frac{4}{3} - 6 \] \[ y = -\frac{4}{3}x - \frac{14}{3} \] So, the equation of the line is: \[ y = -\frac{4}{3}x - \frac{14}{3} \] And voilà! You have the line's equation in slope-intercept form. It’s fascinating to look at the history of coordinate geometry! The ideas of lines and slopes can be traced back to ancient civilizations, but the formalization we know today largely came from René Descartes in the 17th century, who introduced the Cartesian plane. With his innovative thinking, Descartes married algebra and geometry in a way that allowed people to visualize equations as geometric shapes! Did you know that understanding the equation of a line can be incredibly helpful in various real-world scenarios? For instance, if you're an architect, knowing how to find the slope of a roof or determining the optimal angle for sunlight can directly impact energy efficiency. Similarly, in economics, linear equations can model relationships between supply and demand, helping predict market behaviors! It’s all about connecting math to the real world!