he point-slope form of the equation of the line that passes through \( (-9,-2) \) and \( (1,3) \) is \( y-3=\frac{1}{2}(x-1) \). What \( y=\frac{1}{2} x+2 \) \( y=\frac{1}{2} x-4 \) \( y=\frac{1}{2} x+\frac{5}{2} \) \( y=\frac{1}{2} x-\frac{7}{2} \)

To convert the point-slope form \( y - 3 = \frac{1}{2}(x - 1) \) into slope-intercept form \( y = mx + b \), start by distributing \( \frac{1}{2} \) on the right side: \( y - 3 = \frac{1}{2}x - \frac{1}{2} \). Next, add 3 to both sides: \( y = \frac{1}{2}x - \frac{1}{2} + \frac{6}{2} = \frac{1}{2}x + \frac{5}{2} \). So, the correct answer is \( y = \frac{1}{2}x + \frac{5}{2} \). In real-world applications, lines can represent relationships in various fields. For instance, in finance, the point-slope form can be used to depict cost versus production levels. Say a factory has startup costs and then a linear cost for every unit produced, creating a helpful model for budgeting and predicting expenses over time. Moreover, common mistakes occur when students forget to distribute correctly or miscalculate when plugging in point coordinates. Always double-check your arithmetic! Additionally, simplifying too soon can lead to skipped essential steps that help visualize and understand the problem better. Keeping track of each transformation helps avoid confusion and errors.