Using the Point-Slope Form Previously you learned one strategy for writing a linear equation when given the slope and a point on the line. In this lesson you will learn a different strategy – one that uses the point-slope form of an equation of a line. POINT-SLOPE FORM OF THE EQUATION OF A LINE The point-slope form of the equation of the nonvertical line that passes through a given point (x 1, y 1) with a slope of m is y – y 1 = m (x – x 1).

Developing the Point-Slope Form Write an equation of the line. Use the points (2, 5) and (x, y). SOLUTION You are given one point on the line. Let (x, y) be any point on the line. Because (2, 5) and (x, y) are two points on the line, you can write the following expression for the slope of the line. m = y – 5 x – 2 Use formula for slope. The graph shows that the slope is . Substitute for m in the formula for slope. 2 3 y – 5 x – 2 = 2 3 Substitute for m. 2 3 y – 5 = 2 3 (x – 2) Multiply each side by (x – 2). 2 3 The equation y – 5 = (x – 2) is written in point-slope form.

Using the Point-Slope Form You can use the point-slope form when you are given the slope and a point on the line. In the point-slope form, (x 1, y 1) is the given point and (x, y) is any other point on the line. You can also use the point-slope form when you are given two points on the line. First find the slope. Then use either given point as (x 1, y 1).

Using the Point-Slope Form Write an equation of the line shown below. SOLUTION First find the slope. Use the points (x 1, y 1) = (–3, 6) and (x 2, y 2) = (1, –2). m = y 2 – y 1 x 2 – x 1 –2 – 6 1 – (–3) = –8 4 = = –2

Using the Point-Slope Form Write an equation of the line shown below. SOLUTION Then use the slope to write the point-slope form. Choose either point as (x 1, y 1). y – y 1 = m (x – x 1) Write point-slope form. y – 6 = –2[x – (–3)] Substitute for m, x 1 and y 1. y – 6 = –2(x + 3) Simplify. y – 6 = –2 x – 6 Use distributive property. y = –2 x Add 6 to each side.

Modeling a Real-Life Situation MARATHON The information below was taken from an article that appeared in a newspaper. Use the information to write a linear model for optimal running pace, then use the model to find the optimal running pace for a temperature of 80°F.

Writing and Using a Linear Model SOLUTION Let T represent the temperature in degrees Fahrenheit. Let P represent the optimal pace in feet per second. From the article, you know that the optimal running pace at 60°F is 17.6 feet per second so one point on the line is (T 1, P 1) = (60, 17.6). Find the slope of the line. m = – 0.3 5 change in P change in T = = – 0.06 Use the point-slope form to write the model. P – P 1 = m (T – T 1) Write the point-slope form. P – 17.6 = (– 0.06)(T – 60) Substitute for m, T1, and P1. P – 17.6 = – 0.06T + 3.6 Use distributive property. P = –0.06T + 21.2 Add 17.6 to each side. †

Writing and Using a Linear Model SOLUTION Use the model P = –0.06T + 21.2 to find the optimal pace at 80°F. P = – 0.06(80) + 21.2 = 16.4 At 80°F the optimal running pace is 16.4 feet per second. CHECK A graph can help you check this result. You can see that as the temperature increases the optimal running pace decreases.