1. Direct
and
Inverse
VARIATION
Section 8.1

2. a is called the constant of variation.
We will do an example together.

4. 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 = ax
24 = a•3
a = 8
Write the general equation
Substitute

5. (b) Find y when x=2
First we find the constant of variation, which was a = 8
Now we substitute into y = ax.
y = ax
y = 8•2
y = 16

6. Another method of solving direct variation problems is to use proportions.
So lets look at a problem that can by solved by either of these two methods.

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

8. Now lets solve using the equation.
y = ax
y = ax
Either method gives the correct answer, choose the easiest for you.

9. 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

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

12. Write an inverse variation equation
The variables x and y vary inversely and y=7 when x = 4. Write an equation that relates x and y. Then find y when x = -2.

13. Find y when x=15, if y varies inversely as x and x=10 when y=12
Solve by equation:

14. A general form of the proportion
Lets do an example that can be solved by using the equation and the proportion.

15. Solve by proportion:
Find y when x=15, if y varies inversely as x and x=10 when y=12

16. Try it yourself!!
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

17. Review of Variations Direct Variation Inverse Variation Formula
General Equation

19. Inverse or Direct??

20. Hint: Solve the equation for y and take notice of the relationship.
Ex: tell whether x & y show direct variation, inverse variation, or neither.
xy=4.8
y=x+4 c.
Inverse Variation
Hint: Solve the equation for y and take notice of the relationship.
Neither
Direct Variation

23. Joint Variation
When a quantity varies directly as the product of 2 or more other quantities.
For example: if z varies jointly with x & y, then z=axy.
How would you write the following?
If y varies jointly with x and z, then
If x varies jointly with y and z, then

25. Example

26. Solve for y
If y varies jointly as x and z, and y = 54 when x = 2 and z = 9, find y when x = 7 and z = 10.
Since a = 3, then y = axz
y = 3•7•10
y = 210

28. Try it yourself!!
If y varies jointly as x and z, and y = 48 when x = 3 and z = 4, find the constant of variation.
If x varies jointly as y and z, and x = 36 when y = 36 and z = 4, find x when y = 12 and z = 8.
a = 4
x = 24