3-6

1. 3-6 Lines in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

2. Objectives Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding.

3. Vocabulary point-slope form slope-intercept form

5. Example 1A: Writing Equations In Lines Write the equation of each line in the given form. the line with slope 6 through (3, –4) in point-slope form Point-slope form y – y1 = m(x – x1) y – (–4) = 6(x – 3) Substitute 6 for m, 3 for x1, and -4 for y1.

6. Example 1B: Writing Equations In Lines Write the equation of each line in the given form. the line through (–1, 0) and (1, 2) in slope-intercept form Find the slope. Slope-intercept form y = mx + b 0 = 1(-1) + b Substitute 1 for m, -1 for x, and 0 for y. 1 = b Write in slope-intercept form using m = 1 and b = 1. y = x + 1

7. 5 5 Substitute for m, 3 for x1, and 0 for y1. y – 0 = (x – 3) 3 3 5 y = (x - 3) 3 Example 1C: Writing Equations In Lines Write the equation of each line in the given form. the line with the x-intercept 3 and y-intercept –5 in point slope form Use the point (3,-5) to find the slope. y – y1 = m(x – x1) Point-slope form Simplify.

8. Check It Out! Example 1a Write the equation of each line in the given form. the line with slope 0 through (4, 6) in slope-intercept form Point-slope form y – y1 = m(x – x1) Substitute 0 for m, 4 for x1, and 6 for y1. y – 6 = 0(x – 4) y = 6

9. Check It Out! Example 1b Write the equation of each line in the given form. the line through (–3, 2) and (1, 2) in point-slope form Find the slope. y – y1 = m(x – x1) Point-slope form Substitute 0 for m, 1 for x1, and 2 for y1. y – 2 = 0(x – 1) y - 2 = 0 Simplify.

10. The equation is given in the slope-intercept form, with a slope of and a y-intercept of 1. Plot the point (0, 1) and then rise 1 and run 2 to find another point. Draw the line containing the points. run 2 rise 1 (0, 1) Example 2A: Graphing Lines Graph each line.

11. The equation is given in the point-slope form, with a slope of through the point (–4, 3). Plot the point (–4, 3) and then rise –2 and run 1 to find another point. Draw the line containing the points. rise –2 (–4, 3) run 1 Example 2B: Graphing Lines Graph each line. y – 3 = –2(x + 4)

12. (0, –3) Example 2C: Graphing Lines Graph each line. y = –3 The equation is given in the form of a horizontal line with a y-intercept of –3. The equation tells you that the y-coordinate of every point on the line is –3. Draw the horizontal line through (0, –3).

13. The equation is given in the slope-intercept form, with a slope of and a y-intercept of –3. Plot the point (0, –3) and then rise 2 and run 1 to find another point. Draw the line containing the points. run 1 rise 2 (0, –3) Check It Out! Example 2a Graph each line. y = 2x – 3

14. The equation is given in the point-slope form, with a slope of through the point (–2, 1). Plot the point (–2, 1)and then rise –2 and run 3 to find another point. Draw the line containing the points. rise –2 (–2, 1) run 3 Check It Out! Example 2b Graph each line.

15. (0, –4) Check It Out! Example 2c Graph each line. y = –4 The equation is given in the form of a horizontal line with a y-intercept of –4. The equation tells you that the y-coordinate of every point on the line is –4. Draw the horizontal line through (0, –4).

17. Example 3A: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. y = 3x + 7, y = –3x – 4 The lines have different slopes, so they intersect.

18. Both lines have a slope of , and the y-intercepts are different. So the lines are parallel. Example 3B: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. Solve the second equation for y to find the slope-intercept form. 6y = –2x + 12

19. Example 3C: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. 2y – 4x = 16, y – 10 = 2(x - 1) Solve both equations for y to find the slope-intercept form. 2y – 4x = 16 y – 10 = 2(x – 1) 2y = 4x + 16 y – 10 = 2x - 2 y = 2x + 8 y = 2x + 8 Both lines have a slope of 2 and a y-intercept of 8, so they coincide.

20. Check It Out! Example 3 Determine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide. Solve both equations for y to find the slope-intercept form. 3x + 5y = 2 3x + 6 = –5y 5y = –3x + 2 Both lines have the same slopes but different y-intercepts, so the lines are parallel.

21. Example 4: Problem-Solving Application Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same?

22. 1 Understand the Problem The answer is the number of miles for which the costs of the two plans would be the same. Plan A costs $100.00 for the initial fee and $0.35 per mile. Plan B costs $85.00 for the initial fee and $0.50 per mile.

23. Make a Plan 2 Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations.

24. 3 Solve 0 = –0.15x + 15 Plan A: y = 0.35x + 100 Plan B: y = 0.50x + 85 Subtract the second equation from the first. x = 100 Solve for x. Substitute 100 for x in the first equation. y = 0.50(100) + 85 = 135

25. 3 Solve Continued The lines cross at (100, 135). Both plans cost $135 for 100 miles.

26. Look Back 4 Check your answer for each plan in the original problem. For 100 miles, Plan A costs $100.00 + $0.35(100) = $100 + $35 = $135.00. Plan B costs $85.00 + $0.50(100) = $85 + $50 = $135, so the plans cost the same.

27. 2. the line through (5, –1) with slope in point-slope form. y + 1= (x – 5) 2 5 Lesson Quiz: Part I Write the equation of each line in the given form. Then graph each line. 1. the line through (-1, 3) and (3, -5) in slope-intercept form. y = –2x + 1

28. 1 2 Lesson Quiz: Part II Determine whether the lines are parallel, intersect, or coincide. 3. y – 3 = – x, y – 5 = 2(x+ 3) intersect 4.2y = 4x + 12, 4x – 2y = 8 parallel