y = kx with k ¹ 0.
k is called the constant of variation
or constant of proportionality.
During a thunderstorm, Peter
recorded how long it was
between seeing the lightning
and hearing the thunder. The
distance d in kilometers
between Peter and the
lightning can be estimated by
d=1/3s, where s is the number
of seconds between seeing
the lightning and hearing the
thunder.
The table below shows Peter’s
recorded data. Notice how, as
time increases, the distance
also increases.
Time (sec)
Distance(km)
2
4
6
8
12
0.6
1.3
2.0
2.6
4
If we graph
on a coordinate plane we get…
Looking at the graph, what is the
slope of the line?
Answer: 1/3
Looking at the equation, what is
the constant of variation?
Answer: 1/3
The constant of variation and the
slope are the same!!!!
y varies inversely as x if k ¹ 0
k
such that xy=k or y = x
For k being the constant of
variation or proportionality
Graphically an inverse variation is
a curve, not a straight line as in
the direct variation.
The graph shows
the number of hours,
y,
it takes x workers
to paint a room.
If y varies directly as x, and y=24 and x=3 find:
(a) the constant of variation
Write the general equation
Substitute
Solve
24 = k × 3
k =8
y = kx
(b) Find y when x=2
We found k in part (a): k=8
Now we substitute the value of k into
y=kx to get the direct variation formula:
y=8x
Finally, plug in the given value and solve
to find the missing variable:
y = 8× 2
y =16
Another method of solving direct
variation problems is to use
proportions.
y1 y2
=
x1 x2
So lets look at a problem that can
by solved by either of these two
methods.
If y varies directly as x and y=6 when x=5,
then find y when x=15.
Proportion Method:
6 y
=
1. Plug in the given values 5 15
2. Solve for the missing variable
5y = 90
y = 18
Now lets solve using the equation.
y = kx
6 = k×5
6
k=
5
y = kx
6
y = ×15
5
y =18
Either method gives the correct
answer, choose the easiest for you.
Now you do one on your own.
y varies directly as x, and x=8 when
y=9. Find y when x=12.
Answer: 13.5
Applications of Direct Variation
According to Hook’s Law, the
force F required to stretch a spring
x units beyond its natural length
varies directly as x. A force of 30
pounds stretches a certain spring 5
inches. Find how far the spring is
stretched by a 50 pound weight.
F1 F2
=
x1 x2
30 50
=
5
x
30x = 250
1
x = 8 inches
3
Set up a proportion
Substitute
Now try this problem.
Use Hook’s Law to find how many
pounds of force are needed to
stretch a spring 15 inches if it takes
18 pounds to stretch it 13.5 inches.
Answer: 20 pounds
If y varies inversely as x, and y=12 when x=10
find:
(a) the constant of variation
Write the general equation
Substitute
Solve
k
12 =
10
k =120
k
y=
x
(b) Find y when x=15
We found k in part (a): k=120
k
y=
x
Now we substitute the value of k into
to get the direct variation formula: y = 120
x
Finally, plug in the given value and solve
to find the missing variable:
120
y=
15
y=8
Solve by proportion:
y1 y2
=
x2 x1
15 10
=
12 y
15y =120
y =8
Solve this problem using either
method.
Find x when y=27, if y varies
inversely as x and x=9 when
y=45.
Answer: 15
Applications of Inverse Variation
The pressure P of a compressed
gas is inversely proportional to its
volume V according to Boyle’s
Law. A pressure of 40 pounds per
square inch is created by 600
cubic inches of a certain gas. Find
the pressure when the gas is
compressed to 200 cubic inches.
y1 y2
=
x2 x1
40
x
=
200 600
200x = 24000
x = 120 pounds / in
2
Now try this one on your own.
A pressure of 20 pounds per inch
squared is exerted by 400 inches
cubed of a certain gas. Use Boyle’s
Law to find the pressure of the gas
when it is compressed to a volume
of 100 inches cubed.
Answer: 80 pounds / in
2
Solving variation word problems.
1. Understand the problem.
2. Write the formula.
3. Identify the known values and substitute in
the formula.
4. Solve for the unknown.
5. Make sure to include units of measure
Direct Variation Summary
y=k × x with k ¹ 0.
y
or k =
x
 Linear




There is no adding or subtracting
y is directly proportional to x
y varies directly with x
Constant of proportionality is k
Direct Variation
from a Table
To determine direct variation from a table
use the form
y
k=
x
If y divided by x equals the same number
throughout the table, that is direct variation
and that value is k
Inverse Variation Summary
k
y= with k ¹ 0.
x





or k = x × y
Curve
There is no adding or subtracting
y is inversely proportional to x
y varies inversely with x
Constant of proportionality is k
Inverse Variation
from a Table
To determine direct variation from a table
use the form
k = x×y
If x times y equals the same number
throughout the table, that is inverse
variation and that value is k