4.7 Writing Equations in Point-Slope Form How can you write an equation of a line when you are given the slope and a point on the line? 1 ACTIVITY: Writing Equations of Lines Work with a partner. ● Sketch the line that has the given slope and passes through the given point. ● Find the y-intercept of the line. ● Write an equation of the line. 1 3 a. m = −2 b. m = — y y 8 6 7 5 6 4 5 3 4 2 3 1 2 Ź5 Ź4 Ź3 Ź2 Ź1 Ź1 1 Ź6 Ź5 Ź4 Ź3 Ź2 Ź1 Ź1 1 2 3 4 5 Writing Equations In this lesson, you will ● write equations of lines using a slope and a point. ● write equations of lines using two points. Preparing for Standard 8.F.4 184 Chapter 4 3 4 5 6 7 x 1 2 3 4 5 6 x Ź3 Ź2 Ź4 Ź3 Ź5 Ź4 Ź6 5 2 c. m = −— COMMON CORE 2 Ź2 6 x 2 3 1 d. m = — y y 6 7 5 6 4 5 3 4 2 3 1 2 Ź5 Ź4 Ź3 Ź2 Ź1 Ź1 1 2 3 4 5 6 7 x 1 Ź2 Ź6 Ź5 Ź4 Ź3 Ź2 Ź1 Ź1 Ź3 Ź2 Ź4 Ź3 Ź5 Ź4 Ź6 Ź5 Graphing and Writing Linear Equations 2 ACTIVITY: Deriving an Equation Work with a partner. y a. Draw a nonvertical line that passes through the point (x1, y1). (x1, y1) b. Plot another point on your line. Label this point as (x, y). This point represents any other point on the line. Math Practice Construct Arguments O x c. Label the rise and the run of the line through the points (x1, y1) and (x, y). How does a graph help you derive an equation? d. The rise can be written as y − y1. The run can be written as x − x1. Explain why this is true. e. Write an equation for the slope m of the line using the expressions from part (d). f. 3 Multiply each side of the equation by the expression in the denominator. Write your result. What does this result represent? ACTIVITY: Writing an Equation Work with a partner. Savings Account For 4 months, you saved $25 a month. You now have $175 in your savings account. A 250 ● Draw a graph that shows the balance in your account after t months. Use your result from Activity 2 to write an equation that represents the balance A after t months. Balance (dollars) 225 ● 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 t Time (months) 4. Redo Activity 1 using the equation you found in Activity 2. Compare the results. What do you notice? 5. Why do you think y − y1 = m(x − x1) is called the point-slope form of the equation of a line? Why do you think it is important? 6. IN YOUR OWN WORDS How can you write an equation of a line when you are given the slope and a point on the line? Give an example that is different from those in Activity 1. Use what you learned about writing equations using a slope and a point to complete Exercises 3 – 5 on page 188. Section 4.7 Writing Equations in Point-Slope Form 185 4.7 Lesson Lesson Tutorials Key Vocabulary point-slope form, p. 186 Point-Slope Form A linear equation written in the form y − y1 = m(x − x1) is in point-slope form. The line passes through the point (x1, y1), and the slope of the line is m. Words y slope (x, y) y Ź y1 y − y1 = m(x − x1) Algebra (x1, y1) x Ź x1 passes through (x1, y1) EXAMPLE 1 O x Writing an Equation Using a Slope and a Point Write in point-slope form an equation of the line that passes through 2 3 the point (−6, 1) with slope —. y − y1 = m(x − x1) Write the point-slope form. 2 3 Substitute — for m, −6 for x1, and 1 for y1. 2 3 Simplify. 2 3 y − 1 = —[x − (−6)] y − 1 = —(x + 6) 2 3 So, the equation is y − 1 = — (x + 6). Check Check that (−6, 1) is a solution of the equation. 2 3 y − 1 = —(x + 6) Write the equation. ? 2 1 − 1 = —(−6 + 6) Substitute. 3 0=0 Exercises 6 – 11 Chapter 4 Simplify. Write in point-slope form an equation of the line that passes through the given point and has the given slope. 1. (1, 2); m = −4 186 ✓ 2. (7, 0); m = 1 Graphing and Writing Linear Equations 3. 3 4 (−8, −5); m = −— EXAMPLE 2 Writing an Equation Using Two Points Write in slope-intercept form an equation of the line that passes through the points (2, 4) and (5, −2). Study Tip y −y x2 − x1 You can use either of the given points to write the equation of the line. Use m = −2 and (5, −2). y − (−2) = −2(x − 5) y + 2 = −2x + 10 y = −2x + 8 −6 3 Then use the slope m = −2 and the point (2, 4) to write an equation of the line. ✓ y − y1 = m(x − x1) Write the point-slope form. y − 4 = −2(x − 2) Substitute −2 for m, 2 for x1, and 4 for y1. y − 4 = −2x + 4 Distributive Property y = −2x + 8 EXAMPLE −2 − 4 5−2 2 1 Find the slope: m = — = — = — = −2 3 Write in slope-intercept form. Real-Life Application You finish parasailing and are being pulled back to the boat. After 2 seconds, you are 25 feet above the boat. (a) Write and graph an equation that represents your height y (in feet) above the boat after x seconds. (b) At what height were you parasailing? a. You are being pulled down at the rate of 10 feet per second. So, the slope is −10. You are 25 feet above the boat after 2 seconds. So, the line passes through (2, 25). Use the point-slope form. 10 feet per second y − 25 = −10(x − 2) Substitute for m, x1, and y1. y − 25 = −10x + 20 Distributive Property y = −10x + 45 Write in slope-intercept form. So, the equation is y = −10x + 45. b. You start descending when x = 0. The y-intercept is 45. So, you were parasailing at a height of 45 feet. y 45 y â Ź10x à 45 40 35 30 (2, 25) 25 20 15 10 5 0 Exercises 12 – 17 0 1 2 3 4 5 6 7 x Write in slope-intercept form an equation of the line that passes through the given points. 4. (−2, 1), (3, −4) 5. (−5, −5), (−3, 3) 6. (−8, 6), (−2, 9) 7. WHAT IF? In Example 3, you are 35 feet above the boat after 2 seconds. Write and graph an equation that represents your height y (in feet) above the boat after x seconds. Section 4.7 Writing Equations in Point-Slope Form 187 Exercises 4.7 Help with Homework 1. VOCABULARY From the equation y − 3 = −2(x + 1), identify the slope and a point on the line. 2. WRITING Describe how to write an equation of a line using (a) its slope and a point on the line and (b) two points on the line. 6)=3 9+(- 3)= 3+(- 9)= 4+(- = 1) 9+(- Use the point-slope form to write an equation of the line with the given slope that passes through the given point. 1 2 3 4 3. m = — 4. m = −— 5. m = −3 y y 4 y 4 3 3 3 2 2 2 1 1 1 Ź4 Ź3 Ź2 Ź1 Ź1 1 2 Ź5 Ź4 Ź3 Ź2 Ź1 Ź1 3 x 1 2 x Ź2 Ź1 Ź1 1 2 3 4 5 x Ź2 Ź2 Ź2 Ź3 Ź3 Ź3 Ź4 Write in point-slope form an equation of the line that passes through the given point and has the given slope. 1 2 3 3 4 6. (3, 0); m = −— 7. (4, 8); m = — 1 7 5 3 9. (7, −5); m = −— 10. (3, 3); m = — 8. (1, −3); m = 4 11. (−1, −4); m = −2 Write in slope-intercept form an equation of the line that passes through the given points. 2 12. (−1, −1), (1, 5) 15. (4, 1), (8, 2) 13. (2, 4), (3, 6) 14. (−2, 3), (2, 7) 16. (−9, 5), (−3, 3) 17. (1, 2), (−2, −1) 18. CHEMISTRY At 0 °C, the volume of a gas is 22 liters. For each degree the temperature T (in degrees Celsius) increases, the volume V (in liters) of the 2 25 gas increases by —. Write an equation that represents the volume of the gas in terms of the temperature. 188 Chapter 4 Graphing and Writing Linear Equations 19. CARS After it is purchased, the value of a new car decreases $4000 each year. After 3 years, the car is worth $18,000. a. Write an equation that represents the value V (in dollars) of the car x years after it is purchased. b. What was the original value of the car? 20. REASONING Write an equation of a line that passes through the point (8, 2) that is (a) parallel and (b) perpendicular to the graph of the equation y = 4x − 3. 21. CRICKETS According to Dolbear’s law, you can predict the temperature T (in degrees Fahrenheit) by counting the number x of chirps made by a snowy tree cricket in 1 minute. For each rise in temperature of 0.25°F, the cricket makes an additional chirp each minute. a. A cricket chirps 40 times in 1 minute when the temperature is 50°F. Write an equation that represents the temperature in terms of the number of chirps in 1 minute. b. You count 100 chirps in 1 minute. What is the temperature? c. The temperature is 96 °F. How many chirps would you expect the cricket to make? Leaning Tower of Pisa y 22. WATERING CAN You water the plants in your classroom at a constant rate. After 5 seconds, your watering can contains 58 ounces of water. Fifteen seconds later, the can contains 28 ounces of water. (10.75, 42) a. Write an equation that represents the amount y (in ounces) of water in the can after x seconds. b. How much water was in the can when you started watering the plants? c. When is the watering can empty? 23. Problem The Leaning Tower of Pisa in Italy was built Solving between 1173 and 1350. a. Write an equation for the yellow line. x 7.75 m b. The tower is 56 meters tall. How far off center is the top of the tower? Graph the linear equation. (Section 4.4) 24. y = 4x 25. y = −2x + 1 26. y = 3x − 5 27. MULTIPLE CHOICE What is the x-intercept of the equation 3x + 5y = 30? (Section 4.5) A −10 ○ B −6 ○ Section 4.7 C 6 ○ D 10 ○ Writing Equations in Point-Slope Form 189

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