Direct, Inverse, and Joint Variation
Direct Variation:
y varies directly as x
Described by the eq’n:
So
y  kx
y
k
x
where k is the constant of variation
Note: A way to remember is Direct and
Divide both start with the letter D.
To solve, set up your problem as follows:
y1 y2

x1 x2
Inverse Variation:
y varies inversely as x
Described by the eq’n:
So
y
k
x
k  yx
where k is the constant of variation
Note: A way to remember is inverse is the
opposite of direct so if you divide for direct,
then you must multiply for the inverse.
To solve, set up your problem as follows:
y1 x1  y2 x2
Joint Variation:
y varies jointly as x and z
Described by the eq’n:
So
y  kxz
y
k
xz
where k is the constant of variation
Note: Joint variation works the same way as
direct variation.
To solve, set up your problem as follows:
y1
y
 2
x1 z1 x2 z2
Examples:
1. If y varies directly as x and y = 6 when x = 11, find y when x = 3, find y.
2. Find x when y = 8 if x varies directly as y and y = 2 when x = 5.
3. If y varies inversely as x and y = 5 when x = 2 find y when x = 12.
4. Find y when x = 18 if x varies inversely as y and y = 3 when x = 4.
5. y varies inversely as x and directly as z. y = 20 when x =5 and z = 2. Find z
when y = 1- and x = 30.
6. y varies directly as x and inversely as z, y = 3 when x = 6 and z = -4. Find z
when y = -5 and x = 10.
7. Find y when x = 4 and z = 15, if y varies jointly as x and z and y = 5 when z =
8 and x = 10.
8. Find y when x = 12 and z = 2. if x varies jointly as y and z and y = 24 when z
=3 and x =1.
9. z is inversely proportional to the square of y and directly proportional to x.
Solve for x.
10. Volume (V) of a gas is directly proportional to Temperature (T) and inversely
proportional to the pressure (P). Solve for P.