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Lesson 5.1
Direct Variation
Direct Variation:
𝒚
𝒙
= k or 𝒚 = 𝒌𝒙 or 𝒚 = 𝒎𝒙
Direct variation is a relationship between two variables in
which one variable is a constant multiple of the other. The
rate of change is constant (constant of variation).
A direct variation is a situation in which two quantities, such
as hours and pay, increase or decrease at the same rate.
That is as one quantity doubles, the other quantity also
doubles. The variables are said to be directly proportional.
Direct Variation:
𝒚
𝒙
= k or 𝒚 = 𝒌𝒙 or 𝒚 = 𝒎𝒙
• If the quantities x and y are related by an equation y = kx or
y=mx for some constant k ≠ 0, we say that:
• y varies directly with x.
• y is directly proportional to x.
• The constant k/m is also called the constant of
proportionality.
Direct Variation:
𝒚
𝒙
= k or 𝒚 = 𝒌𝒙 or 𝒚 = 𝒎𝒙
For example: If you get paid hourly, when you work twice as
many hours, you make twice as much money. How much
you make is a constant multiple of how many hours you
work.
If I make $20/hour, how much will I make after 2 hours?
After 3 hours? After 4 hours?
Direct Variation:
𝒚
𝒙
= k or 𝒚 = 𝒌𝒙 or 𝒚 = 𝒎𝒙
Ex. the relationship between distance and time
• Direct variation occurs when the dependent variable varies
by the same factor as the independent variable.
• If the car travels 50km/h, how far will the car travel after 2
hours? After 3 hours?
Key Concepts:
The equation of a Direct Variation is always in the the form y = mx or
y=kx where m/k is the constant of variation.
• The graph of y = mx is a straight line with the slope of m.
• The line y = mx always passes through the 𝑶𝑹𝑰𝑮𝑰𝑵 (0, 0).
• (0, 0) is called the origin.
• The relation y = mx represents direct variation because there is a direct
relationship.
• Constant of variation: in a direct variation, the ratio of corresponding
values of the variables often represented by 𝒌 or 𝒎, or the constant
multiple by which one variable is multiplied.
Let’s look at the following example:
Jim drives at a constant speed of 100km/h. How far will Jim
drive in 8 hours?
1) Create a table showing the distance travelled in 0 hours, 1 hour, 2 hours & so
on up to 8 hours.
2) Identify the independent variable (x axis) and the dependent variable (y axis)
3) Graph this relationship.
4) Describe the shape of the graph? Where does it intersect the vertical axis?
5) Declare variables and write an equation to find the distance 𝑑 in km that Jim
drives in 𝑡 hours.
6) Use the equation to determine the distance travelled by Jim in 12.5 hours.
Time (hours)
Distance (km)
0
0
Independent variable:
Dependent variable:
Distance vs time
900
800
Distance (km)
700
600
500
400
300
200
100
0
0
1
2
3
4
Time (hours)
5
6
7
8
9
• This graph shows a straight line that slopes upward from left to right
and intersects the vertical axis at the origin (0,0).
𝑑 = 100𝑡
𝑑 = 100 12.5
𝑑 = 1 250𝑘𝑚
Consider the distance Jim drove in 2 hours.
What happens to the distance when the time is doubled?
What happens to the distance when the time is tripled?
In this example distance varies directly with time. This
relation is known as a direct variation.
In a direct variation, the ratio of corresponding values of the
variables is a constant.
𝒅
𝒕
So if d is distance and t is time, then = 𝐤 or 𝒅 = 𝒌𝒕 where
𝒌 is called the constant of variation.
Examples:
1) v = 3t the constant of variation is 3
2) d = 20t the constant of variation is 20
3) y = 6x the constant of variation is 6
4) The distance, d, in kilometers, varies directly with the time, t, in
hours.
𝑑 = 𝑘𝑡
a) Find the constant of variation if d = 42 when t = 0.5. Rearrange the
equation relating d and t. (Find 𝑘)
b) How long will it take to travel a distance of 252 km? Rearrange the
equation for 𝑡.
Another example:
Use a table to organize the information in the following
problem:
How long does it take to fill a 3 000 litre hot tub if a water truck
supplies water at a rate of 250 litres per hour and the tub is
initially empty?
Time (h)
Water (l)
What if the water truck was
able to double the flow rate?
Time (h)
Water (l)
• Which variable is dependent? (If a variable is dependent its
value cannot be known without knowing the other variable’s
value). To determine the dependent variable ask yourself the
following key questions:
Does the amount of water depend on time?
Or
Does time DEPEND on the amount of water ?
Jack works part-time at the Christmas market. He earns $8.5/h.
a) Describe the relationship between his pay, in dollars, and the
time, in hours, he works – declare variables .
b) Represent this relationship with an equation. What is the
constant of variation?
c) One week, Jack works 9 hours. Find his pay for that week.
Ben makes Christmas baskets to sell at the market. He earns
$8/h. How long must he work to earn $150?
a) Write an equation to represent the relationship between
Ben’s pay and the hours he works – declare variables .
b) Rearrange the equation to solve for ℎ.
Homework:
p. 242 – 245 C1 # 1-6, 8, 9, 12 a- d