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(8) Lesson 3.3
1. Course 3, Lesson 3-3
Find the slope of the line that passes through each pair of points.
1. A(0, 0), B(4, 3) 2. M(–3, 2), N(7, –5)
3. P(–6, –9), Q(2, 7) 4. K(6, –3), L(16, –4)
5. Do the following points form a parallelogram when they are
connected? Explain. (Hint: Two lines that are parallel have the same slope.)
A(5, 4), B(10, 4), C(5, –1), D(0, 0)
6. What is the slope of the graph at the right?
2. Course 3, Lesson 3-3
ANSWERS
1.
2.
3. 2
4.
5. no; the slope of is 0, but the slope of is . Therefore,
is not parallel to .
6.
3
4
7
10
1
10
AB DC
1
5
AB DC
1
3
3. WHY are graphs helpful?
Expressions and Equations
Course 3, Lesson 3-3
7. To
• write and solve direct variation
equations,
• compare direct variations by using
equations or graphs
Course 3, Lesson 3-3
Expressions and Equations
8. • direct variation
• constant of variation
• constant of proportionality
Course 3, Lesson 3-3
Expressions and Equations
9. Course 3, Lesson 3-3
Expressions and Equations
Words A linear relationship is a direct
variation when the ratio of y to x is
a constant, m. We say y varies
directly with x.
Symbols m = or y = mx, where m is the
constant of variation and m 0
Example y = 3x
y
x
10. 1
Need Another Example?
2
3
Step-by-Step Example
1. The amount of money Robin earns while
babysitting varies directly with the time as
shown in the graph. Determine the amount
that Robin earns per hour.
To determine the amount Robin
earns per hour, or the unit rate,
find the constant of variation.
Use the points (2, 15), (3, 22.5), and (4, 30).
So, Robin earned $7.50 for each hour she babysits.
11. Answer
Need Another Example?
The amount of money Serena
earns at her job is shown on the
graph. Determine the amount
Serena earns per hour.
$10 per hour
12. 1
Need Another Example?
2
3
Step-by-Step Example
2. A cyclist can ride 3 miles in 0.25 hour. Assume that the distance
biked in miles y varies directly with time in hours x. This situation
can be represented by y = 12x. Graph the equation. How far can
the cyclist ride per hour?
Make a table of values. Then graph the equation y = 12x.
In a direct variation equation, m represents the slope. So,
the slope of the line is .
The unit rate is the slope of the line. So, the cyclist can ride
12 miles per hour.
13. Answer
Need Another Example?
Some types of bamboo can grow 7 inches in 3.5 hours.
Assume that the height y varies directly with the time x. This
situation can be represented by the equation y = 2x. Graph
the equation. How fast can the bamboo grow per hour?
2 inches per hour
14. Course 3, Lesson 3-3
Expressions and Equations
You can use tables, graphs, words, or equations to represent and compare
proportional relationships.
Words y varies directly with x
Equation
1
5
y x
Table Graph
15. 1
Need Another Example?
2
3
Step-by-Step Example
3. The distance y in miles covered
by a rabbit in x hours can be
represented by the equation
y = 35x. The distance covered
by a grizzly bear is shown on
the graph. Which animal is
faster? Explain.
The slope or unit rate is 35 mph.
Since 35 > 30, the rabbit is the faster animal.
Rabbit y = 35x
Grizzly Bear Find the slope of the graph.
1
30
16. Answer
Need Another Example?
Mike spent the amounts shown in the table on
tokens at Playtime Games.
Tokens at Game Time are $0.25 per token. Which
arcade has the best price for tokens? Explain.
Playtime Games; Sample answer: The unit for
Playtime Games is $0.20 per token and the unit
rate for Game Time is $0.25 per token.
17. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
4. A 3-year-old dog is often considered to be 21 in human years.
Assume that the equivalent age in human years y varies directly
with its age as a dog x. Write and solve a direct variation equation
to find the human-year age of a dog that is 6 years old.
So, when a dog is 6 years old, the
equivalent age in human years is 42.
Graph the equation y = 7x.
Let x represent the dog’s actual age and let y represent the human-equivalent age.
You want to know the human-year age or y-value when the dog is 6 years old.
y = mx Direct variation
21 = m(3) y = 21, x = 3
7 = m Simplify.
y = 7x Replace m with 7
y = 7x Write the equation.
y = 7 • 6 x = 6
y = 42 Simplify.
The y-value when x = 6 is 42.
Check
18. Answer
Need Another Example?
At a certain store, four cans of soup cost $5. Assume the
total cost is directly proportional to the number of cans
purchased. Write and solve a direct variation equation to
find how much it would cost to buy 10 cans of soup.
y = 1.25x; $12.50
19. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-3
Expressions and Equations
20. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-3
Expressions and Equations
Sample answers:
• You can use graphs to compare different direct
variation relationships.
• You can find the unit rate in a relationship by looking at
a graph of the relationship.
21. Explain what a constant
of variation is in a
direct variation.
Ratios and Proportional RelationshipsExpressions and Equations
Course 3, Lesson 3-3