2. Concepts and Objectives
Variation
Solve for the constant of variation
Solve direct and inverse variation problems
Solve joint variation problems
3. Direct Variation
y varies directly as x, or y is directly proportional to x,
if there exists a nonzero real number k, called the
constant of variation, such that
y = kx.
Let n be a positive real number. Then y varies directly
as the nth power of x, or y is directly proportional to
the nth power of x, if there exists a nonzero real
number k such that n
y kx
4. Inverse Variation
y varies inversely as x, or y is inversely proportional
to x, if there exists a nonzero real number k, such that
Let n be a positive real number. Then y varies inversely
as the nth power of x, or y is inversely proportional to
the nth power of x, if there exists a nonzero real
number k such that
k
y
x
n
k
y
x
5. Solving Variation Problems
1. Write the general relationship among the variables as
an equation. Use the constant k.
2. Substitute given values of the variables and find the
value of k.
3. Substitute this value and the remaining values into the
original equation and solve for the unknown.
6. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.
7. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.
24 8k
3k
3 12y
36y
7
4
k
28k
28
14
y
2y
8. Joint Variation
Let m and n be real numbers. Then y varies jointly as
the nth power of x and the mth power of z if there exists
a nonzero real number k such that
Note: If n or m is negative, then the variable is said to
vary inversely.
n m
y kx z
9. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.
10. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.
48 12 8k
0.5k
0.5 10 6u
30u
2
6
8
3
k
12k
2
12 10
4
z
7.5z