Question 1: A straight line that passes through \( (1,2) \) and \( (3,6) \). The equation of the line is

To find the equation of the line passing through the points \( (1, 2) \) and \( (3, 6) \), we first calculate the slope \( m \) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in our points, \( m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \). Now, using the point-slope form \( y - y_1 = m(x - x_1) \) with one of the points \( (1, 2) \), we get \( y - 2 = 2(x - 1) \), or simplified, the equation of the line is \( y = 2x \). The concept of finding the slope of a line dates back to ancient civilizations like the Greeks, who were intrigued by geometry. Aristarchus of Samos famously studied proportions in relationships, which is foundational for linear equations today. Fast forward to now, this understanding allows us to solve real-world problems like predicting costs, determining speed, or even analyzing trends in data. Isn’t mathematics timeless? In practice, a common mistake when determining line equations is forgetting to simplify or miscalculating the slope. Always remember to double-check your coordinates and do not rush through the arithmetic! It’s easy to mix up the points or their corresponding y-values, which can lead to a different (and incorrect) line altogether. Take it slow and steady!