The general equation for DIRECT VARIATION is y = kx with k 0.

k is called the constant of variation.

We will do an example together.

If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation

(b) Find y when x=2

(a) Find the constant of variation

y kx Write the general equation

24 k 3 Substitute

k 8

(b) Find y when x=2

First we find the constant of variation, which was k=8

Now we substitute into y=kx.

y kx

y 82y 16

Another method of solving direct variation problems is to use proportions.

If y1 = kx1, then k =y1

x1

and

If y 2 kx2 , then k =y2

x2

Therefore...

y1

x1

y2

x2

So lets look at a problem that can by solved by either of these two methods.

If y varies directly as x and y=6 when x=5, then find y when x=15.

Proportion Method:6

5

y

15Let x1 5, y1 6, x2 15, y2 y

5y 90

y 18

Now lets solve using the equation.

y kx

6 k 5

k 6

5

y kx

y 6

515

y 18

Either method gives the correct answer, choose the easiest for you.

Now you do one on your own.

y varies directly as x, and x=8 when y=9. Find y when x=12.

Answer: 13.5

What does the graph y=kx look like?A straight line with a y-intercept of 0.

5

-5

-10 10

f x = 3 x

Inverse Variation

y varies inversely as x if k 0

such that xy=k or y k

x

Just as with direct variation, a proportion can be set up solve problems of indirect variation.

x1

y2

x2

y1

A general form of the proportion

Lets do an example that can be solved by using the equation and the proportion.

Find y when x=15, if y varies inversely as x and x=10 when y=12

Solve by equation:

xy k

10 12 k

120 k

xy k

15 y 120y 8

Solve by proportion:

x1

y2

x2

y1

15

12

10

y

15y 120

y 8

Solve this problem using either method.

Find x when y=27, if y varies inversely as x and x=9 when y=45.

Answer: 15

Joint Variation

For three quantities x, y and z, if there is a constant k such that

z = kxy

We say “z varies jointly as y and x” or

“z is jointly proportional to x and y”.

A general form of the proportion

Lets do an example that can be solved by using the equation and the proportion.

Example 9

• U varies jointly as V and the square of W. if V =4 and W = 3, then U = 18. Find the value of V when U = 24 and W = 3.

Class work…

• Work book page 123 and 124 even