Question

a line passes through the points (8, 1) and (3, 5). what is its equation in point - slope form? use one of the specified points in your equation. write your answer using integers, proper fractions, and improper fractions. simplify all fractions.

Explanation:

Step1: Find the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(8, 1)\) and \((x_2, y_2)=(3, 5)\). So, \( m=\frac{5 - 1}{3 - 8}=\frac{4}{-5}=-\frac{4}{5} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use either of the two points. Let's use the point \((8, 1)\) (we could also use \((3, 5)\)). Substituting \( m = -\frac{4}{5} \), \( x_1 = 8 \), and \( y_1 = 1 \) into the point - slope formula, we get \( y - 1=-\frac{4}{5}(x - 8) \). If we use the point \((3, 5)\), substituting \( m=-\frac{4}{5}\), \( x_1 = 3 \), and \( y_1 = 5 \) into the formula, we get \( y - 5=-\frac{4}{5}(x - 3) \).

Answer:

\( y - 1=-\frac{4}{5}(x - 8) \) (or \( y - 5=-\frac{4}{5}(x - 3) \))