The document defines direct and inverse variation and provides examples of translating statements describing direct and inverse variation into mathematical equations. It also provides two examples of using the equations to solve for unknown variables given specific values of other variables. Specifically, it defines direct variation as being proportional and inverse variation as being inversely proportional. It translates statements about direct and inverse variation into equations using a constant of variation k. The examples show setting the equations equal to known values and solving for the constant k and then using k to solve for the unknown variable.

1. COMBINED
VARIATION

2. Definition
The statement “z varies directly as x and
inversely as y” means 𝑧 =
𝑘𝑥
𝑦
, or 𝑘 =
𝑧𝑦
𝑥
,
where k is the constant variation.

3. TRANSLATING STATEMENT TO EQUATION
Translating statements into mathematical equations using k as the constant of
variation.
a. T varies directly as a and inversely as b
𝑇 =
𝑘
𝑎
b. Y varies directly as x and inversely as the square of z
𝑌 =
𝑘𝑥
𝑧2

4. Translate each statement into mathematical equation.
1. W varies jointly as c and the square of a and inversely as b
2. P varies directly as the square of x and inversely as s
3. The electrical resistance R of a wire varies directly as its length and
inversely as the square of its diameter d
4. The acceleration A of a moving object varies directly as the
distance d it travels and inversely as the square of the time t it
travels
5. The pressure P of a gas varies directly as its temperature t and
inversely as the volume V
𝑾 =
𝒌𝒄𝒂𝟐
𝒃
𝑷 =
𝒌𝒙𝟐
𝒔
𝑹 =
𝒌𝒍
𝒅𝟐
𝑨 =
𝒌𝒅
𝒕𝟐
𝑷 =
𝒌𝒕
𝒗
ACTIVITY

5. EXAMPLE 1
If z varies directly as x and inversely as y, and z = 9 when x = 6 and y = 2.
Find z when x =8 and y = 12.
Solution:
𝑧 =
𝑘𝑥
𝑦
9 =
6𝑘
2
9 = 3𝑘
𝑘 = 3
𝑧 =
3𝑥
𝑦

6. EXAMPLE 1
If z varies inversely as x and inversely as y, and z = 9 when x = 6 and y = 2.
Find z when x =8 and y = 12.
Solution:
𝑧 =
3𝑥
𝑦
𝑧 =
3(8)
12
𝑧 =
24
12
𝑧 = 2

7. EXAMPLE 2
If s varies directly as r and inversely as t, and s=15 when r=20 and t=40, find s
when r=12 and t=20.
Solution:
𝑠 =
𝑘𝑟
𝑡
𝑘 =
𝑠𝑡
𝑟
𝑘 =
(15)(40)
(20)
𝑘 =
600
20
𝑘 = 30
𝑠 =
30(12)
20
𝑠 =
360
20
𝑠 = 18