The document discusses the concept of joint variation, where a variable z varies jointly with variables x and y according to an equation of the form z = kxy, where k is a constant of variation. It provides examples of finding the constant of variation and equation of variation given values for z, x, and y. It also gives an example of using the equation of variation to determine the value of one variable when values are given for the other variables.

2. Let’s Learn
The simple interest earned in a given time depends on two
quantities, the principal and the rate. If both r either of the two
quantities increases, what do you think will happen to the
interest? What will be the effect of a decrease in the principal
and/or the rate?
The formula for the simple interest earned in a given time is
l = Prt, where t constant. This shows that interest varies jointly
with the principal and the rate.

3. Consider also, the area (A) of a triangle with base (b) and height (h).
A =
1
2
bh
We say that the area of the triangle varies jointly as the height and
the base. In the above illustration, the constant of variation is
1
2
.
An equation of the form z = kxy, k ≠ 0, where k is constant,
expresses joint variation. This equation means that z varies jointly
with x and y.

4. Example 1. Suppose z = 42 when x = 3 and y = 6. Find the constant of
variation and the equation of variation if z varies jointly as x and y.
Steps Solution
1. Identify the given z = 42 when x = 3 and y = 6
z varies jointly as x and y
2. Determined what is asked for. Find the constant of variation and
the equation of variation
3. Identify the formula to be used z = kxy

5. Steps Solution
4. Substitute the given in the
formula to solve the constant of
variation
z = kxy
42 = k(3)(6)
42 = 18k
42
18
=
7
3
= k
5. Find the equation of variation z = kxy
z =
7
3
xy
6. State the answer. The constant of variation is
𝟕
𝟑
and
the equation of variation is z=
𝟕
𝟑
xy.

6. Example 2. Suppose c = 48 when a = 12 and b = 2. Find the constant of
variation and an equation of variation where c varies jointly as a and b.
Solution
Given:
c = 48 when a = 12 and b = 2
c varies jointly as a and b
Asked:
Find the constant of variation and an
equation of variation
c = kab
c = kab
48 = k(12)(2)
48 = 24k
2 = k (constant of variation)
c = kab
c = 2ab (equation of variation)
The constant of variation is 2 and
the equation of variation is c= kab.

7. Example 3. The variable z varies jointly as the variable x and y.When x = 7
and y = 2, z = 42. What is the value off z when x = 9 and y = 5?
Solution
Given:
x = 7 and y = 2, z = 42
z varies jointly as the variable x and y
Asked:
Find the value of z when x = 9 and y = 5.
z = kxy
42 = k(7)(2)
42 = 14k
3 = k (constant of variation)
Find the value of z when x = 9 and y = 5.
z = kxy
z = 3(9)(5)
z = 135
The value of z = 135 when x = 9 and
y = 5.

8. Example 4. If P varies jointly as Q and the square of R, and P = 200 when
Q = 2 and R = 5, what is R when P = 128 and Q = 2?
Solution
P = kQR2
200 = k (2)(5)2
200 = k (2)(25)
200= 50k
4 = k
P = kQR2
128 = (4)(2)R2
128 = 8R2
128
8
= R2
16 = R2
16 = R
4 = R
The value of R is 4 when P = 128
and Q = 2.