1. 3-5 Slopes of Lines Warm Up Lesson Presentation Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Warm Up
Find the value of m.
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3. Objectives Find the slope of a line.
Use slopes to identify parallel and perpendicular lines.

4. Vocabulary
rise
run
slope

5. The slope of a line in a coordinate plane
is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

7. Example 1A: Finding the Slope of a Line
Use the slope formula to determine the slope of each line. AB
Substitute (–2, 7) for (x1, y1) and (3, 7) for (x2, y2) in the slope formula and then simplify.

8. Example 1B: Finding the Slope of a Line
Use the slope formula to determine the slope of each line. AC
Substitute (–2, 7) for (x1, y1) and (4, 2) for (x2, y2) in the slope formula and then simplify.

9. Example 1C: Finding the Slope of a Line
Use the slope formula to determine the slope of each line. AD
Substitute (–2, 7) for (x1, y1) and (–2, 1) for (x2, y2) in the slope formula and then simplify.
The slope is undefined.

10. A fraction with zero in the denominator is undefined because
it is impossible to divide by zero.
Remember!

11. Example 1D: Finding the Slope of a Line
Use the slope formula to determine the slope of each line. CD
Substitute (4, 2) for (x1, y1) and (–2, 1) for (x2, y2) in the slope formula and then simplify.

12. Check It Out! Example 1
Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1).
Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify.

13. One interpretation of slope is a rate of change
One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

14. Example 2: Transportation Application
Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line.
Use the points (4, 260) and (7, 455) to graph the line and find the slope.

15. Example 2 Continued
The slope is 65, which means Justin is traveling at an average of 65 miles per hour.

16. Check It Out! Example 2
What if…? Use the graph below to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same.
Since Tony is traveling at an average speed of 60 miles per hour, by 6:30 P.M. Tony would have traveled 390 miles.

17. What does it mean when lines have equal slopes? Draw an example.

18. How can you tell if lines are perpendicular. Draw an example
How can you tell if lines are perpendicular? Draw an example. Using their slopes?

20. If a line has a slope of , then the slope of a perpendicular line is .
The ratios and are called opposite reciprocals.

21. Four given points do not always determine two lines.
Graph the lines to make sure the points are not collinear.
Caution!

22. Example 3A: Determining Whether Lines Are Parallel, Perpendicular, or Neither
Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.
UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3)
The products of the slopes is –1, so the lines are perpendicular.

23. Example 3B: Determining Whether Lines Are Parallel, Perpendicular, or Neither
Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.
GH and IJ for G(–3, –2),
H(1, 2), I(–2, 4), and J(2, –4)
The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

24. Example 3C: Determining Whether Lines Are Parallel, Perpendicular, or Neither
Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.
CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)
The lines have the same slope, so they are parallel.

25. Check It Out! Example 3a
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3)
Vertical and horizontal lines are perpendicular.

26. Check It Out! Example 3b
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1)
The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

27. Check It Out! Example 3c
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
BC and DE for B(1, 1), C(3, 5), D(–2, –6), and E(3, 4)
The lines have the same slope, so they are parallel.

28. Lesson Quiz
1. Use the slope formula to determine the slope of the line that passes through M(3, 7) and N(–3, 1).
m = 1
Graph each pair of lines. Use slopes to determine whether they are parallel, perpendicular, or neither.
2. AB and XY for A(–2, 5),
B(–3, 1), X(0, –2), and Y(1, 2)
4, 4; parallel
3. MN and ST for M(0, –2),
N(4, –4), S(4, 1), and T(1, –5)

29. The equation of a line can be written in many different forms
The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line.

31. A line with y-intercept b contains the point (0, b).
A line with x-intercept a contains the point (a, 0).
Remember!

32. 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
y – y1 = m(x – x1)
Point-slope form
y – (–4) = 6(x – 3)
Substitute 6 for m, 3 for x1, and -4 for y1.

33. 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

34. 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
Substitute for m, 3 for x1, and 0 for y1. 5 3
y – 0 = (x – 3) 5 3
y = (x - 3) 5 3
Simplify.

35. 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
y – y1 = m(x – x1)
Point-slope form
Substitute 0 for m, 4 for x1, and 6 for y1.
y – 6 = 0(x – 4)
y = 6

36. 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.

37. A system of two linear equations in two variables represents two lines
A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.

39. 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.

40. 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
Both lines have a slope of , and the y-intercepts are different. So the lines are parallel.

41. 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.

42. 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.

43. 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?

44. Understand the Problem1
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 $ for the initial fee and $0.35 per mile. Plan B costs $85.00 for the initial fee and $0.50 per mile.

45. 2
Make a Plan
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.

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

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

48. Check It Out! Example 4
What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan?
The lines would be parallel.

49. 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
2. the line through (5, –1) with slope in point-slope form. 2 5
y + 1 = (x – 5)

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