1. 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. Using the Point-Slope Form 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 ).

2. Multiply each side by (x – 2). Write an equation of the line. Use the points (2, 5) and (x, y). Use formula for slope. SOLUTION You are given one point on the line. Let (x, y) be any point on the line. m =m = y – 5 x – 2 The graph shows that the slope is. Substitute for m in the formula for slope. 2323 2323 y – 5 x – 2 = 2323 Substitute for m. 2323 y – 5 = 2323 (x – 2) 2323 The equation y – 5 = (x – 2) is written in point-slope form. Developing the Point-Slope Form Because (2, 5) and (x, y) are two points on the line, you can write the following expression for the slope of the line.

3. 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 ).

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

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

6. 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. MARATHON The information below was taken from an article that appeared in a newspaper. Modeling a Real-Life Situation

7. Writing and Using a Linear Model SOLUTION Let T represent the temperature in degrees Fahrenheit. 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 = – 0.35– 0.35 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, T 1, and P 1. P – 17.6 = – 0.06T + 3.6 Use distributive property. P = –0.06T + 21.2 Add 17.6 to each side. Let P represent the optimal pace in feet per second.

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