powerpoint presentation

1. combined Variation

2. LET’S LEARN
Suppose z = 100 when w = 4, x =5, and y = 15. If z varies
jointly as w and x and inversely as y, find the equation of
variation. Moreover, solve for z when w = 1, x = 5, and y = 3.
This kind of variation is called combined variation. The
equation of variation is given by
z = k
𝑤𝑥
𝑦

3. A combined variation involves a combination of a
direct or joint variation, and an inverse variation.

4. Solution:
We now solve for k in the equation
when z = 100, w = 4,
x = 5, and y = 15.
z = k
𝑤𝑥
𝑦
100 = k
(4)(5)
15
; 100 = k
20
15
100 =
4
3
k ; 300 = 4k
300
4
= k
75 = k (constant of variation)
Therefore, the equation of combined
variation is z = 75
𝑤𝑥
𝑦
.

5. Finally, we solve fore z when w = 1, x = 5, and y = 3.
z = k
𝑤𝑥
𝑦
z = 75
(1)(5)
3
= 75
5
3
= 125
So, z = 125 when w = 1, x = 5, and y = 3.

6. STUDY MORE EXAMPLES
• 1. Suppose z = 96 when w = 48, x = 6 and y = 4. Find the
constant of variation and the equation of variation if z varies
directly as w and varies inversely as x and y.
Given:
z = 96 when w = 48, x = 6 and y = 4
z varies directly with w
z varies inversely with x and y

7. Asked:
Find the constant of variation and the equation of variation.
Solution:
z = k
𝑤
𝑥𝑦
96 = k
48
(6)(4)
=
48
24
= 2
96 = k(2)
48 = k (constant of variation)
z = 48
𝑤
𝑥𝑦
(equation of variation)

8. 2. The variable z varies directly as the variable w and varies
inversely as x, y when w = 54, x = 9, and y = 24 we have z = 15.
What is the value of z when w = 20, x = 15, and y = 2?
Given:
w = 54, x = 9, y = 24, z = 15
z varies directly with w
z varies inversely with x, y
Asked:
Find the value of z when w = 20, x = 15, and y = 2.

9. Solution
z = k
𝑤
𝑥𝑦
15 = k
54
(9)(24)
=
54
216
=
1
4
15 = k
1
4
or k(0.25)
15 = 0.25k
15
0.25
= k
60 = k (constant of variation)
Find the value of z when w = 20, x =
15, and y = 2.
z = k
𝑤
𝑥𝑦
z = 60
20
(15)(2)
z = 60
20
30
z =
(60)(20)
30
=
1,200
30
= 40
When w = 20, x = 15, and y = 2, z = 40.