Question

the points (2, 7) and (7, -3) fall on a particular line. 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 \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \((2, 7)\) and \((7, -3)\), we have \( x_1 = 2\), \( y_1 = 7\), \( x_2 = 7\), \( y_2=-3 \). So \( m=\frac{-3 - 7}{7 - 2}=\frac{-10}{5}=-2 \).

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 \((2, 7)\) (we could also use \((7,-3)\)). Substituting \( m=-2 \), \( x_1 = 2 \), and \( y_1 = 7 \) into the point - slope formula, we get \( y - 7=-2(x - 2) \). If we use the point \((7,-3)\), substituting \( m = - 2\), \( x_1=7\), and \( y_1=-3\) into the formula, we get \( y+3=-2(x - 7) \).

Answer:

Using the point \((2,7)\): \( y - 7=-2(x - 2) \) or using the point \((7,-3)\): \( y + 3=-2(x - 7) \)