Question

the points (6, 6) and (3, 7) 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: Calculate 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 \((6, 6)\) and \((3, 7)\), we have \( x_1 = 6,y_1 = 6,x_2=3,y_2 = 7 \). So \( m=\frac{7 - 6}{3 - 6}=\frac{1}{-3}=-\frac{1}{3} \).

Step2: Recall point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \((x_1,y_1)\) is a point on the line and \( m \) is the slope. We can use either of the two points. Let's use the point \((6,6)\) first. Substituting \( m = -\frac{1}{3}\), \( x_1 = 6\) and \( y_1 = 6\) into the point - slope formula, we get \( y - 6=-\frac{1}{3}(x - 6) \). We could also use the point \((3,7)\): substituting \( m=-\frac{1}{3}\), \( x_1 = 3\) and \( y_1 = 7\) into the formula, we get \( y - 7=-\frac{1}{3}(x - 3) \).

Answer:

\( y - 6=-\frac{1}{3}(x - 6) \) (or \( y - 7=-\frac{1}{3}(x - 3) \))