1. Variations

2. We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant.
Variations

3. We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations

4. Example A. Translate the following phrases into equations.
a. y varies directly to x.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations

5. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations

6. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz

7. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.

8. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.
c. y varies directly to x2z2

9. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.
c. y varies directly to x2z2
y = kx2z2 for some k.

10. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.
c. y varies directly to x2z2
y = kx2z2 for some k.
d. The cost C varies directly with the square of the length L.

11. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.
c. y varies directly to x2z2
y = kx2z2 for some k.
d. The cost C varies directly with the square of the length L.
The square of the length L is L2.

12. Example A. Translate the following phrases into equations.
a. y varies directly to x.
y = kx for some k.
We say the variable y varies (directly) to an expression f if
y = k·f
where k is a constant. The formula y = k* f is called the
general (direct) variation equation.
Variations
b. y varies directly to xz
y = kxz for some k.
c. y varies directly to x2z2
y = kx2z2 for some k.
d. The cost C varies directly with the square of the length L.
The square of the length L is L2.
Hence the general equation is y = kL2 for some k.

13. We say the variable y varies inversely to an expression f if
y =
where k is a constant.
Variations
k
f

14. We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f

15. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f

16. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
y = where k is a constant

17. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y = where k is a constant

18. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
k
x2z
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y = where k is a constant
y = where k is a constant

19. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
k
x2z
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y = where k is a constant
y = where k is a constant
c. The intensity of light I varies inversely to the square of
distance D

20. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
k
x2z
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y = where k is a constant
y = where k is a constant
c. The intensity of light I varies inversely to the square of
distance D
The square of distance D is D2.

21. Example B. Translate the following phrases into equations.
a. y varies inversely to x.
k
x
k
x2z
We say the variable y varies inversely to an expression f if
y =
where k is a constant. The formula y = is called the
general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y = where k is a constant
y = where k is a constant
c. The intensity of light I varies inversely to the square of
distance D
The square of distance D is D2.
Hence I =
k
D2 where k is a constant.

22. In general, a variation problem gives the type of the variation
and the values of the variables.
Variations

23. In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations

24. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations

25. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.

26. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.
Set y = –6 and x = –4 in the general equation.

27. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.
Set y = –6 and x = –4 in the general equation.
–6 = k(–4)

28. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
k = =
–6
–4 2
3
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.
Set y = –6 and x = –4 in the general equation.
–6 = k(–4) so

29. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
k = =
–6
–4
If we put this into the general equation, we have the
specific equation:
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.
Set y = –6 and x = –4 in the general equation.
–6 = k(–4) so
2
3

30. Example C.
a. Given that y varies directly to x and y = –6 when x = –4.
Find the constant k and the specific variation equation.
k = =
–6
–4
If we put this into the general equation, we have the
specific equation:
In general, a variation problem gives the type of the variation
and the values of the variables. From these, we solve for the
constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y = kx.
Set y = –6 and x = –4 in the general equation.
–6 = k(–4) so
y = x
2
3
2
3

31. Variations
b. Find the value of x if y = 20.

32. Variations
b. Find the value of x if y = 20.
20 = x
2
3
Substitute y = 20 into the specific equation y = x
2
3

33. Variations
b. Find the value of x if y = 20.
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
y = x
2
3

34. b. Find the value of x if y = 20.
Variations
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
So x = 40/3 if y = 20.
y = x
2
3

35. b. Find the value of x if y = 20.
Variations
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
So x = 40/3 if y = 20.
In application, the language of variation sums up the basic
relation of two or more types of variables.
y = x
2
3

36. b. Find the value of x if y = 20.
Variations
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
So x = 40/3 if y = 20.
In application, the language of variation sums up the basic
relation of two or more types of variables.
y = x
2
3
Example D. The weight W of an object
varies inversely to the square of the
distance D from the center of the earth.
A person weighs 160 pounds on the
surface which is 4000 miles to the center of
the earth. What would the person weigh
if he’s 6000 miles above the surface of the
earth?

37. b. Find the value of x if y = 20.
Variations
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
So x = 40/3 if y = 20.
In application, the language of variation sums up the basic
relation of two or more types of variables.
y = x
2
3
4000 m.
W = 160
Example D. The weight W of an object
varies inversely to the square of the
distance D from the center of the earth.
A person weighs 160 pounds on the
surface which is 4000 miles to the center of
the earth. What would the person weigh
if he’s 6000 miles above the surface of the
earth?

38. b. Find the value of x if y = 20.
Variations
20 = x
2
3
Substitute y = 20 into the specific equation
20 * = x
2
3
So x = 40/3 if y = 20.
In application, the language of variation sums up the basic
relation of two or more types of variables.
y = x
2
3
4000 m.
6000 m.
W = 160
W = ?Example D. The weight W of an object
varies inversely to the square of the
distance D from the center of the earth.
A person weighs 160 pounds on the
surface which is 4000 miles to the center of
the earth. What would the person weigh
if he’s 6000 miles above the surface of the
earth?

39. Variations
The square of the distance D is D2.

40. Variations
The square of the distance D is D2.
So the general equation is W = .
D2
k

41. Variations
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

42. Variations
Substitute W = 160, D = 4000 into the specific equation.
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

43. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

44. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
k160 * 16,000,000 =
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

45. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
k160 * 16,000,000 =
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

46. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth.
k160 * 16,000,000 =
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

47. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
k160 * 16,000,000 =
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

48. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
k160 * 16,000,000 =
W = D2
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

49. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
When the person is 6000 miles above the surface D = 10000,
k160 * 16,000,000 =
W = D2
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

50. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
When the person is 6000 miles above the surface D = 10000,
we have
k160 * 16,000,000 =
W = D2
2.56 * 109
W =
(10000)2
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

51. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
When the person is 6000 miles above the surface D = 10000,
we have
k160 * 16,000,000 =
W = D2
2.56 * 109
W =
(10000)2
2.56 * 109
=
108
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .

52. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
When the person is 6000 miles above the surface D = 10000,
we have
k160 * 16,000,000 =
W = D2
2.56 * 109
W =
(10000)2
2.56 * 109
=
108
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .
108
10

53. Variations
160 =
(4000)2
k
Substitute W = 160, D = 4000 into the specific equation.
So k = 2.56 * 109
(This constant is specific to earth. For another planet,
the general equation is the same but k would be different.)
Hence the specific equation is
When the person is 6000 miles above the surface D = 10000,
we have
k160 * 16,000,000 =
W = D2
2.56 * 109
W =
(10000)2
2.56 * 109
=
108
2.56 * 109
The square of the distance D is D2.
So the general equation is W = .
D2
k
We are to find k .
108
10
= 25.6 lb

54. Variations
If the expression f is a product of two or more variables,
we also say y varies jointly to these variables.

55. Variations
If the expression f is a product of two or more variables,
we also say y varies jointly to these variables. Hence
“y varies directly to xz” is the same as
“y varies jointly to x and z”.

56. Variations
If the expression f is a product of two or more variables,
we also say y varies jointly to these variables. Hence
“y varies directly to xz” is the same as
“y varies jointly to x and z”.
“y varies directly to x2z2 ” is the same as
“y varies jointly to x2 and z2”.

57. Variations
Exercise A. Write down the general equations for the
following variation statements.
5. T varies directly to P. 6. T varies inversely to V.
1. R varies directly to D. 2. R varies inversely to T.
7. A varies directly to r2.
9. C varies directly to D3
8. y varies inversely to d2.
3. S varies directly to T. 4. S varies inversely to N.
10. C varies directly to xy.
11. The rate R that a car is traveling at varies directly to the
distance D it covers in a fixed amount of time.
12. The rate R that a car is traveling at varies inversely to the
time T it takes to travel a fixed distance.
Each person in a group of N people has to chip in to buy a large
pizza that cost $T. Let S be the share of each person, then
13. S varies directly to T
14. N varies inversely to S.

58. Variations
16. The temperature varies inversely to the volume.
17. The area of a circle varies directly to the square of its
radius.
18. The light–intensity varies inversely to the square of the
distance.
19. The mass required varies inversely to the square of the
speed it was traveling.
20. The cost of cheese balls varies directly to the 3rd power
(cube) of the diameter of the ball.
22. The time seemly has passed doing something varies
inversely to how much fun there was doing it.
21. The fun of doing something varies inversely to how
much time seemly has passed doing it.
15. The temperature varies directly to the pressure.
Assign variables and write down the variation equations.

59. Variations
24. The light–intensity L underwater varies inversely to the
square of the depth of the distance. If at 3 feet the intensity L
is 5 ft–candle, find the specific equation and what is the
intensity at the depth of 10 ft.
25. The price of a pizza varies directly to the square of the
diameter of the pizza. Given that a 6” pizza is $5.50, find
the specific equation for the price in terms of the diameter.
What should be the price of a 12” pizza?
26. The cost of a solid chocolate ball varies directly to the
cube of its diameter. A chocolate ball with 3” diameter cost
$25, find the specific equation of the cost in terms of the
diameter and how much is one with a 1– foot diameter?
23. The number of marbles N that we are able to buy varies
inversely to the price P of a marble. We’re able to buy 30
marbles at $45 each. What is the specific equation of the N in
terms of P and what is the price P if we are only able to buy
10 marbles?