through \( (1,-3) \) and \( (4,-2) \) A. Write the point-slope form of the equation of the line through the given points: B. \( y=-\frac{1}{3}(x+1) \) B. \( y+1=\frac{1}{3}(x+3) \) C. \( y-1=-\frac{1}{3}(x+3) \) D. \( y+3=\frac{1}{3}(x-1) \)

To find the point-slope form of the equation of a line through the points \( (1, -3) \) and \( (4, -2) \), we first need to calculate the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( y_2 = -2 \), \( y_1 = -3 \), \( x_2 = 4 \), and \( x_1 = 1 \). So, the slope \( m = \frac{-2 - (-3)}{4 - 1} = \frac{1}{3} \). Now we can write the point-slope form using the slope and one of the given points, say \( (1, -3) \): \[ y - (-3) = \frac{1}{3}(x - 1) \] or \[ y + 3 = \frac{1}{3}(x - 1) \] Alright, so regarding that given line \(y + 3 = \frac{1}{3}(x - 1)\), it looks very similar to what we calculated! However, out of the choices provided, we should select the option that matches our derived line equation: A. \( y + 3 = \frac{1}{3}(x - 1) \) is the correct point-slope form! Now, let’s look at real-world applications and pitfalls to avoid when working with linear equations! Lines are everywhere in our daily lives! For instance, they can represent trends in data – like the relationship between advertising expenses and sales revenue, helping businesses forecast outcomes. Understanding how to represent relationships with equations helps in fields like economics, science, and even architecture! Common mistakes include using the wrong slope or point when writing your equation. Also, mixing up the x and y intervals can significantly change your line’s position on the graph. Always double-check calculations and ensure you're working with the correct points to avoid a mathematical mess!