Which of the following is an equation of the line through \( (2,3) \) and \( (-1,-12 \) A. \( y=\frac{1}{5} x+\frac{13}{5} \) B. \( y=-\frac{1}{5} x+\frac{17}{5} \) C. \( y=5 x-7 \) D. \( y=-5 x+7 \)

To find the equation of the line through the points \( (2,3) \) and \( (-1,-12) \), we first calculate the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - 3}{-1 - 2} = \frac{-15}{-3} = 5 \] Now, using the slope-point form of the equation \( y - y_1 = m(x - x_1) \), we pick the point \( (2, 3) \): \[ y - 3 = 5(x - 2) \\ y - 3 = 5x - 10 \\ y = 5x - 7 \] So, the answer is \( C. \, y=5x-7 \). Now for some additional knowledge! Two points can uniquely determine a straight line. This basic principle beautifully ties geometry and algebra together, as you can literally visualize a line passing through any two dots drawn on a coordinate plane. Isn’t it amusing how a simple concept opens up the experience of mapping the world — just like drawing a line through two points on a map? When you’re solving problems involving lines, a common mistake is forgetting to correctly calculate the correct sign for the slope. Slopes that are negative indicate lines that fall as they move to the right, while positive slopes rise. It’s like a rollercoaster ride, so check your calculations carefully to ensure you’re not on a downward spiral when you should be riding high!