h

1. • illustrate situations that involves a joint variation. the
following variations: (a) joint ; (b) combined.
• translate into variation statement a relationship between
two quantities
• Solve for the missing variables in a given joint variation

2. If z varies jointly as x and y, then
z= kxy
or equivalently
𝒁
𝒙𝒚
= 𝒌.
The constant k is called the constant of
variation.

3. Y varies directly to
two or more
quantities

4. A= 𝒌𝒃𝒄𝟑
A= kbh

5. Z varies jointly with the square
of x and cube of y.
M varies jointly with s and p.

6. Find k.
y =kxz
y=6 x=10 z=8
y= kxz
6 = k(10)(8) 6 =80k k= 6
80

7. Find y. x= 4 z=40
y= kxz
y = (4)(40)
y =
1
(3)(4)
k =
3
40
( )
40
3
y = 12

8. Find k.
y=3 x=2 z=24
𝑧 = 𝑘𝑥2
𝑦
24= (2) (3)k 24=12k k = 2
2
𝑧 = 𝑘𝑥2
𝑦

9. Find z.
x= 3 y=5
z = (2) (3)
z =
2
(2)(9)(5)
k = 2
z = 90
𝑧 = 𝑘𝑥2
𝑦
(5)

10. The variable z varies
jointly with x and y.
Also, z = -75 when x
=3 and y = -5. Then
find z when x = 2 and
y = 6.

11. The area of a rectangle
varies jointly as the length
and the width, and whose A
= 72 sq. cm when l = 12 cm
and w = 2cm. Find the are
of the rectangle whose
length is 15cm and width is
3cm.

12. Find an equation of
variation in which y
varies jointly as x and z
and y = 45 when x = 18
and z = 10. Find y when
x = 20 and x = 30.

13. Page 95

16. Illustrate situations that involves
combined variation.
Translate variation statement into a
relationship involving combined
variation between two quantities .
Solve for the missing variables in a given
combined variation.

17. The idea of joint variation can be extended because in many situations direct and inverse
variations are combined in the same equations.
If z varies directly as x and inversely as y. The equation is
𝒛 =
𝒌𝒙
𝒚
 If z varies jointly as x and y and inversely as t . The equation is
𝒛 =
𝒌𝒙𝒚
𝒕

18. 𝑝 =
𝑘𝑥𝑦
𝑧
𝑎 =
𝑘𝑝
𝑞

19. 𝑎 =
𝑘𝑏𝑐
𝑑2

20. Find k.
150 =
𝑘(5)
2
z=150 x=5 y=2 𝒛 =
𝒌𝒙
𝒚
5k= 300 k =
300
5
k =60

21. Find z.
x= 6 y=10
k =60
𝑧 =
𝑘𝑥
𝑦
𝑧 =
(60)(6)
10
𝑧 =
360
10
𝑧 = 36

22. Suppose f varies
directly as g and f
caries inversely as h.
Find g when f = 18
and h = - 3, if g = 24
when h – 2 and f = 6.

23. Y varies directly as x
and inversely as the
square of z. y = 20
when x = 50 and z = 5.
Find y when x = 3 and
z = 6.

24. If z varies directly as x
and inversely as the
square of y. If z = 4
when x = 200 and y =
5, find z when x = 80
and y = 2.

25. E varies jointly as p
and q and inversely as
the square of r. If e =
18 when p = 8 and q =
12 and r = 4. Find e
when p = 9, q = 10 and
r = 3.

26. Express each formula in words. In each
formula, k is the constant of variation.
Equation/
Formula
Words
1. 𝐿 = 𝑘𝑚𝑛
1. 𝐸 = 𝑘𝑎𝑏2
1. 𝑀 = 𝑘𝑎2
𝑟
1. 𝑁 = 𝑘𝑎2𝑏2
1. 𝑅 = 𝑘𝑎√𝑑