Find an equation for the line that passes through the points \( (3,-6) \) and \( (-5,6) \).

To find the equation of the line that passes through the points \( (3,-6) \) and \( (-5,6) \), we first need to calculate the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-6)}{-5 - 3} = \frac{6 + 6}{-8} = \frac{12}{-8} = -\frac{3}{2}. \] Now, we can use the point-slope form of the equation of a line, which is given by \( y - y_1 = m(x - x_1) \). Using one of our points, say \( (3,-6) \): \[ y - (-6) = -\frac{3}{2}(x - 3). \] Simplifying this gives: \[ y + 6 = -\frac{3}{2}x + \frac{9}{2}. \] Now, subtract \( 6 \) from both sides: \[ y = -\frac{3}{2}x + \frac{9}{2} - 6 \implies y = -\frac{3}{2}x - \frac{3}{2}. \] Thus, the equation of the line is: \[ y = -\frac{3}{2}x - \frac{3}{2}. \] Isn't it interesting how the slope gives you insight into how steep the line will be? In this case, the negative slope indicates that as you move to the right along the x-axis, the line will go down! Such lines can often represent real-world situations like how quickly something depreciates over time or how temperature drops as you ascend in altitude! If you're interested in visualizing this line or experimenting with changing points, graphing tools like Desmos or GeoGebra are fabulous options! They allow you to see how changing coordinates affects the slope, intercepts, and overall behavior of the line, making math truly come alive!