z = kxy z = -12 z = kxy z = -84 z = kxy z = -21 4. a. Combined means together as a whole. b. Combined variation is when a quantity varies jointly with respect to the product of two or more variables. c. The mathematical statement that represents combined variation is a = k(bc) where a varies jointly as b and c multiplied together.

1. A. PreliminariesDirections: If the statement “y varies jointly with respect to x
and z” and the equation is in the form “y = kxz” (where k is
the constant), translate each statement into a mathematical
sentence using this pattern.
1. s varies jointly as r and t
2. V varies jointly as l, w, and h
3. N varies jointly as 𝑀1 and 𝑀2
4. A varies jointly as b and the square of c
5. The electrical voltage V varies jointly as the current I and
the resistance R.

2. Joint
Variations

3. Joint Variation is the same as direct
variation with two or more
variables. The statement “a varies
jointly as b and c” means a = kbc, or
𝒌 =
𝒂
𝒃𝒄
where k is the constant of
variation.

4. Illustrative Example 1:
The pressure P of gas varies jointly
as its density D, and its absolute
temperature T.
Solutions:
P = kDT where k is the constant of
variation

5. 2. Find an equation of variation where a varies
jointly as b and c, and a = 24 when b = 2 and c =
3.
Solutions:
A = kbc
24 = k(2)(3)
24 = 6k
k = 4 Therefore, a = 4bc is the required
equation of variation.

6. 3. 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 area of
the rectangle whose length is 15 cm and whose
width is 3 cm.
Solutions: A = klw
72 = k(12)(2)
72 = 24k
k = 3
Therefore, when l = 15 cm
and w = 3 cm,
A = klw
A = 3(15)(3)
A = 135 sq.cm

7. a. What is joint
variation?

8. b. How do we transform
mathematical
statement in joint
variation equation?

9. Directions: Solve for the value of the constant k of
variation, then find the missing value.
1. If y varies jointly as the product of x and z, and y =
105 when x = 5 and z = 7, find y when x = 9 and z =
10.
2. If y varies jointly as the product of x and z, and y =
1000 when x = 10 and z = 20, find y when x = 8 and
z = 10.
3. A varies jointly with l and w, when A = 24, l = 3 and
w = 2. Find A when l = 12 and w = 7

10. 1.y=270
2.y=360
3.A=336

11. Exercise #2.__. Directions: Solve the
following and BOX your final answer
1. y varies jointly as x and z, find y
if x =3, k = 6 and z = 9
2. c varies jointly as a and b. If c =
45 when a = 15 and b = 3, find c
when a = 21 and b = 8.

12. 1. m varies jointly as n and p. If p =
4 when m = 72 and n = 2, find p
when m = 12 and n = 8.
2. q varies jointly as g and the
square of b. If q = 105 when g =
14 and b = 5, find q when g = 10
and b = 14
3.
4.

13. 1. y =162
2. c = 168
3. p =
𝟏
𝟔
4. q = 588

14. Joint Variation is the same as direct
variation with two or more
variables. The statement “a varies
jointly as b and c” means a = kbc, or
𝒌 =
𝒂
𝒃𝒄
where k is the constant of
variation.

15. Assignment # 2. Solve:
1. z varies jointly with x and y. If x=2, y=3
and z=4, write the variation equation and
find z when x=−6 and y=2.
2. z varies jointly with x and y. If x=5, y=−1
and z=10, write the variation equation and
find z when x=−12 and y=7.

16. 3. z varies jointly with x and y. If x=7, y=3 and
z=−14, write the variation equation and find z
when x=8 and y=3
4.Study
a. What do you mean by the word combined?
b. What is combined variation?
c. What mathematical statement represents
combined variation?

17. 1.
2.
3.