Find the equation of the line that passes through the pair of points. Write your answer in slope intercept form. \( (-3,1),(3,3) \) \( y= \)

To find the equation of the line that passes through the points (-3, 1) and (3, 3), we first determine the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (-3, 1) \) and \( (x_2, y_2) = (3, 3) \). Calculating the slope, we have: \[ m = \frac{3 - 1}{3 - (-3)} = \frac{2}{6} = \frac{1}{3} \] Now, using the slope-intercept form \( y = mx + b \), we can plug in one of our points to find b. Let's use the point (3, 3): \[ 3 = \frac{1}{3}(3) + b \] \[ 3 = 1 + b \] \[ b = 2 \] Thus, the equation of the line in slope-intercept form is: \[ y = \frac{1}{3}x + 2 \] --- The concept of slope is deeply rooted in history, tracing back to the ancient Greeks who used geometry to explore the relationships between points and lines. The formal definition of slope—rise over run—boomed in the 17th century, during the age of analytical geometry when mathematicians like René Descartes laid down the groundwork for linking algebra and geometry. Knowing how to apply the equation of a line is essential in various fields! For instance, architects and engineers use linear equations to determine the angles and slopes of structures. Graphs representing relationships in economics, physics, and biology often rely on equations of lines to model trends, making it a crucial tool across disciplines!