1) Joint variation is a direct relationship between three or more quantities that can be expressed as an equation in the form y=kxz, where y varies jointly with x and z. 2) To find the constant k, set up the equation y/xz = k. Then substitute known values for x, y, and z to solve for k. 3) Examples of joint variation problems include finding the equation when area A varies jointly with base b and height h (A=kbh) or volume V varies jointly with length l, width w, and height h (V=klwh). The constant k can be used to solve other problems involving the same quantities.

1. JOINT
VARIATION
DEFINITION
FORMULA
PROBLEM
SOLVING

2. WHAT IS
JOINT
VARIATIO

3. JOINT VARIATION
Joint variation is a direct relationship between three or more
quantities
The statement “y varies jointly as x and z” is translated as
y=kxz
To find the value of k k=
𝑦
𝑥𝑧
:
𝑦
𝑥𝑧
=
𝑘𝑥𝑧
𝑥𝑧
k=
𝒚

4. TRANSLATING
JOINT
VARIATION
STATEMENT
INTO

5. TRANSLATING JOINT VARIATION
STATEMENT INTO
MATHEMATICAL STATEMENT
EXAMPLE:
1)The area A of a parallelogram varies
jointly as the base b and attitude h:
A= kbh

6. TRANSLATING JOINT VARIATION
STATEMENT INTO
MATHEMATICAL STATEMENT
EXAMPLE:
2) V varies jointly with l, w and h:
V=klwh

7. TRANSLATING JOINT VARIATION
STATEMENT INTO
MATHEMATICAL STATEMENT
EXAMPLE:
3) The heat h produced by an electrical lamp
varies jointly as the resistance r and the
square of the current l:
H = 𝑘𝑟𝑙2

8. SOLVING
PROBLEMS
INVOLVING
JOINT

9. SOLVING PROBLEMS
INVOLVING JOINT
VARIATION1) Find the equation of variation where a varies jointly as b and
a=36 when b=3 and c=4
Solution: a=kbc k=
𝑎
𝑏𝑐
k=
36
3(4)
k=3
A=3bc (equation of variation)
a=kbc
36
(3)(4)
=
𝑘 (3)(4)
(3)(4)

10. SOLVING PROBLEMS
INVOLVING JOINT VARIATION
2) Z varies jointly as x and y and z=10 when x=5 and y=6 find
z when x=7 and y=6
Solution: z=kxy k=
𝑧
𝑥𝑦
k=
60
5(6)
k=2 z=2xy
z=2(7)(6) z=84
𝑧
𝑥𝑦
=
𝑘𝑥𝑦
𝑥𝑦
z=kxy
60
(5𝑥6)
=
𝑘(5)(6)
(5)(6)

11. SOLVING PROBLEMS
INVOLVING JOINT VARIATION
3) The weight w of a cylindrical metal varies jointly as its length l and
the square of the diameter d of its base.
a. If w=6 kg when l=6 cm and d=3 cm find the equation of variation.
b. Find l when w=10 kg and d=2 cm.
c. Find w when d=6 cm and l=1.4 cm w=6 l=6 d=3 cm
W=𝑘𝑙𝑑2