Mastering Direct Variation Problems | Essential Guide & Practice Exercises

1. Section 4.7 Straight to the Point: Direct Variation

2. Questions • How do we identify direct variation from a graph, table, or a point? • How do we write models for direct variation problems? • How do we solve direct variation problems?

3. Direct Variation When the quotient of two variables is equal to a constant each variable is said to vary directlywith the other, or the variables are said to be proportional to each other. The nonzero constant k that is this quotient is called the constant of variationor the constant of proportionality. If one of the variables increases/decreases by some factor, then the other variable increases/decreases by the same factor.

4. Direct Variation Equation If there is direct variation between x and y, then we can write: where k is the constant of variation

5. Recognizing Direct Variation From a Table The quotient will be the same for each point. From a Graph The graph is that of a line that passes through the origin.

6. Exercise 1 Determine whether y varies directly with x. If it does, state the constant of variation and give an equation relating x and y. (a)(b)

7. Exercise 1 Determine whether y varies directly with x. If it does, state the constant of variation and give an equation relating x and y. (a)(b) Yes: k = 3.5; y = 3.5x No

8. Exercise 2 Determine whether y varies directly with x. If it does, state the constant of variation and give an equation relating x and y. (a)(b)

9. Exercise 2 Determine whether y varies directly with x. If it does, state the constant of variation and give an equation relating x and y. (a)(b) Yes: k = 2/3 y = (2/3)x No

10. Direct Variation Problems We can solve direct variation problems in two ways: • Solve for the constant of variation, set up the equation, and use it to solve the problem. • Set up a proportion.

11. For Example Suppose y varies directly with x and that y = 8 when x = 2. Find y when x = 4. OR

12. Exercise 3 Jacen’s pay varies directly with the number of hours he works. When he works 20 hours he makes $240. How much does he make if he works 32 hours? What is the constant of variation in this scenario and what does it represent?

13. Exercise 3 Jacen’s pay varies directly with the number of hours he works. When he works 20 hours he makes $240. How much does he make if he works 32 hours? What is the constant of variation in this scenario and what does it represent? $384 k = $12 per hour, and it is his hourly salary

14. Joint Variation Direct variation involving three or more variables is called joint variation. So if z varies directly with x and y, we have: Problems involving such scenarios can be solved in the same two ways as two variable direct variation.

15. Exercise 4 Suppose M varies jointly with a and b, and that M = 102 when a = 2 and b = 3. Find M when a andb are both 3.

16. Exercise 4 Suppose M varies jointly with a and b, and that M = 102 when a = 2 and b = 3. Find M when a andb are both 3. 153