Lesson 2.5 Writing Equations of Lines

1. Lesson 2.5 Writing Equations of Lines

2. Keyword: point-slope form

5. Example 1 SOLUTION 3 From the graph, you can see that the slope is . Because the line intersects the y-axis at , the y-intercept is m = 5 ( ) – 0, 2 – 2. b = Write an Equation Given Slope and y-Intercept Write an equation of the line shown.

6. Example 1 m b y Use slope-intercept form. x + = 3 3 ( ) – 2 Substitute for m and for b. x y + – 2 = 5 5 3 x 2 y – Simplify. = 5 Write an Equation Given Slope and y-Intercept Use slope-intercept form to write an equation of the line.

7. Checkpoint 1 1 ANSWER y 3x + 1 = 2 2 – 2. – – ANSWER y 2x – 4 m 2, b 4 = = = , – y 3. x – 5 ANSWER = 5 m m 3, b = = = Write an Equation Given Slope and y-Intercept Write an equation of the line that has the given slope and y-intercept. 1. b 1 =

8. Write an equation of the line that passes through and has a slope of . 1 – 4 ( ( ) ) 4, 4, 2 2 SOLUTION Because you know the slope and a point on the line, use point-slope form with and m . 1 – = = ( ) x1,y1 4 ( ) Use point-slope form. – – y y1 m x x1 = 1 1 ( ) – Substitute 2 for y1, for m, and 4 for x1. – – y 2 x 4 – = 4 4 Example 2 Write an Equation Given Slope and a Point

9. 1 – ( ) Point-slope form – – y 2 x 4 = 4 1 – – Distributive property y 2 x 1 + = 4 1 – Slope-intercept form y x 3 + = 4 Example 2 Write an Equation Given Slope and a Point Rewrite the equation in slope-intercept form.

10. 1 ( ) 4, 2 – 4 Example 2 Write an Equation Given Slope and a Point CHECK You can check the result by graphing the equation. Draw a line that passes through and has a slope of . Notice that the line has a y-intercept of 3, which agrees with the slope-intercept form found above.

11. Checkpoint 4. ( ) 3 m 2 = , – – – – 3 2, 3, 1, ANSWER y 2x + 5 = – ( y ) 3x + 7 = ANSWER 5. ( ) – – 4, 5 m 3 , = 2 2 ANSWER y 5x + 14 = m = 3 3 , 6. ( ) 4 m 5 = , 7. – y x 1 ANSWER = Write an Equation Given Slope and a Point Write an equation of the line that passes through the given point and has the given slope. Write your equation in slope-intercept form.

12. Checkpoint ( ) 8, 0 1 1 – 8 , 8. – – m y x + 2 ANSWER = = 4 4 , 3 3 ANSWER 9. 5 5 ( ) 10, m = – y x 14 = Write an Equation Given Slope and a Point Write an equation of the line that passes through the given point and has the given slope. Write your equation in slope-intercept form.

13. Write an equation of the line that passes through and is parallel to the line . – y + 2 x 2 = SOLUTION ( ( ) ) – m 2 4 4 = – – 1, 1, The given line has a slope of 2. Any line parallel to this line will also have a slope of 2. Use point-slope form with and to write an equation of the line. – – y 2x + 5 = – ( ) x1,y1 = ( ) Use point-slope form. – – y y1 m x x1 = Substitute 4 for y1, 2 for m, and 1 for x1. – – [ ] ( ) – – – – y 4 2 x 1 = – – 2 x 2 – y 4 = Distributive property Slope-intercept form Example 3 Write Equations of Parallel Lines

14. Write an equation of the line that passes through and is perpendicular to the line . = SOLUTION ( ( ) ) 4 4 – – 1, 1, The given line has a slope of 2. The slope of any line perpendicular to this line will be the negative reciprocal of 2, which is . Use point-slope form with and to write an equation of the line. – – y 2x + 5 = 1 – ( ) x1,y1 2 1 ( ) – x x1 m = 2 Use point-slope form. – y y1 m = Example 4 Write Equations of Perpendicular Lines

15. 1 Substitute 4 for y1, for m, and 1 for x1. – 2 – y 4 Distributive property = + Slope-intercept form 1 1 9 1 x x 2 2 2 2 – y 4 y = + ( ) – 1 1 [ ] – x = 2 Example 4 Write Equations of Perpendicular Lines

16. Writing Equations of Parallel and Perpendicular Lines Checkpoint – y x + 2 = ( ) 3 ANSWER – – – – – 5 2, 6, 6, 5 ( ( ) ) 11. passes through ; perpendicular to – y x + 2 = – – y x 11 = ANSWER y 4x + 11 = 12. passes through ; parallel to y x + 1 = ANSWER – y 4x 1 = Write an equation of the line described. 10. passes through ; parallel to

17. Writing Equations of Parallel and Perpendicular Lines Checkpoint 13. passes through ; perpendicular to ( ) 3 1 – y x + ANSWER = – 2, 4 5 2 – y 4x 1 = Write an equation of the line described.

19. Example 5 SOLUTION The average rate of change in the wild population is m – 138 6 12. = = – 2006 1995 Write a Linear Model Condors In 1995 there were only 6 California condors living in the wild (all in California). In 2006 there were 138 California condors living in the wild (in California, Arizona, and Mexico). Write a linear model that represents the number of California condors living in the wild between 1995 and 2006. The average rate of change is the slope in a linear model.

20. Example 5 Number in 1995 Average rate of change Years since 1995 Number of condors VERBAL MODEL = + • LABELS Number of condors y (condors) = Number in 1995 6 (condors) = Average rate of change 12 (condors per year) = Years since 1995 t (years) = ALGEBRAIC MODEL y 6 + 12t = Write a Linear Model

21. Example 5 ANSWER y 6 + 12t = A linear model for the number of condors is . Write a Linear Model

22. GRIZZLY BEARS once ranged from Texas to North Dakota and westward to Washington and California. Hunting and habitat loss eliminated grizzlies from 98% of their original U.S. range outside of Alaska by the 1970s.

23. Grizzly Bears In 1979, 13 female grizzly bears with first-year cubs were observed in the Greater Yellowstone Ecosystem (GYE). In 2004, the number observed was 49. Use the verbal model shown to write a linear model for the observed number of females with first-year cubs in the GYE from 1979 to 2004.

25. Slip

26. Homework