Find an equation of the line passing through the pair of points. \[ (4,-1),(-16,14) \] Sketch the line.

To find the equation of the line passing through the points \( (4, -1) \) and \( (-16, 14) \), we first need to calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in our points: \[ m = \frac{14 - (-1)}{-16 - 4} = \frac{14 + 1}{-20} = \frac{15}{-20} = -\frac{3}{4} \] Next, we can use the point-slope form of the line's equation, \( y - y_1 = m(x - x_1) \). Using point \( (4, -1) \): \[ y - (-1) = -\frac{3}{4}(x - 4) \] This simplifies to: \[ y + 1 = -\frac{3}{4}x + 3 \] \[ y = -\frac{3}{4}x + 2 \] Thus, the equation of the line in slope-intercept form is: \[ y = -\frac{3}{4}x + 2 \] Now, sketching the line, you would plot the points \( (4, -1) \) and \( (-16, 14) \), and then draw a straight line through them. Remember to mark the y-intercept at \( (0, 2) \) for clarity! The graph will show a downward slope, reflecting the negative slope value. Enjoy your sketch! - **Historical Note**: The study of linear equations dates back to ancient civilizations, including the Babylonians and Greeks. They were among the first to study and solve equations resembling what we use today. Their legacy shapes modern algebra, making it easier to describe relationships in nearly every field, from physics to economics! - **Real-World Application**: Linear equations like this one are everywhere in our everyday lives! For example, when budgeting, you can represent your expenses and income with a line, helping you visualize your financial health. Understanding lines and slopes can also help in designing products or structures that need to meet specific criteria, such as maintaining slants or angles for stability.