1. Direct Variation
Chapter 5.2

2. Coefficient of Variation
Direct Variation
Equation of Direct Variation
y = kx
y = kx
Coefficient of Variation

3. Direct Variation
When you see: “y varies direcly as x” write
y = kx

4. y = -3x y = kx y = kx Direct Variation
Write the equation of direct variation that has a equal to -3
Replace k with -3
coefficient of variation
y = -3x
y = kx
y = kx

5. y = -3 y = ½(-6) y = kx y = ½x Direct Variation
Assume If the coefficient of variation is ½, then what is the value of y when x = -6?
Replace k with ½
Replace x with -6
Multiply ½ times -6 to get answer
y varies directly as x.
y = -3
y = ½(-6)
y = kx
y = ½x

6. 3 6 = 4x y = 4x y = kx = x 2 4 4 Direct Variation
Assume y varies directly as x. If the coefficient of variation is 4, then what is the value of x when y = 6?
Replace k with 4 and y with 6
Divide 4 on both sides
Reduce the fraction 3
6 = 4x
y = 4x
y = kx
= x 2

7. Direct Variation
Suppose y varies directly as x. If y = 3 when x = 15, then find x when y = 5.
Suppose y varies directly as x. Find x when y = 10 if y = -7 when x = -14.
Suppose y varies directly as x. If x = 15 when y = 12, find x when y = 21.
Suppose y varies directly as x. If x = 24 when y = 8, then what is the coefficient of variation?

8. Direct Variation Word Problems
The amount of a person’s paycheck varies directly with the number of hours worked. For 13 hours of work, the paycheck is $ Find the pay for 25 hours of work.
“varies directly” means:
Remember, “y” always varies directly as “x”
y = kx

9. Direct Variation Word Problems
The amount of a person’s varies directly with the number of For 13 hours of work, the paycheck is $ Find the pay for 25 hours of work.
If “y” varies directly as “x” than what comes before varies directly is y and what comes after it is x
paycheck
hours
worked.

10. Direct Variation Word Problems
The amount of a person’s paycheck varies directly with the number of hours worked. For 13 hours of work, the paycheck is $ Find the pay for 25 hours of work.
Replace paycheck with $73.45 and hours with 13
Paycheck = k(hours)
73.45 = k(13)

11. Direct Variation Word Problems
Divide both sides by 13
Plug k back into the equation
73.45 = k(13)
5.65 = k
y = 5.65x

12. Direct Variation Word Problems
The amount of a person’s paycheck varies directly with the number of hours worked. For 13 hours of work, the paycheck is $ Find the pay for 25 hours of work.
The question asks for how much you’d make in 25 hours, so replace x with 25
Multiply to get your answer
y =
y = 5.65x
y = 5.65(25)

13. Direct Variation Word Problems
The amount of a person’s paycheck varies directly with the number of hours worked. For 13 hours of work, the paycheck is $ Find the pay for 25 hours of work.
K = 5.65 which is how much this person makes per hour.
For 25 hours, this person makes $141.25
y = 5.65x
y =

14. Direct Variation Word Problems
The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.
Directly proportional means direct variation
y = kx

15. Direct Variation Word Problems
The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.
Y varies directly as x, so calories is y and amount of milk is x
y = kx
calories = k(amt of milk)

16. Direct Variation Word Problems
The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.
Calories is 160 and the amount of milk is 8
calories = k(amt of milk)
160 = k(8)

17. Direct Variation Word Problems
160 = k(8)
20 = k
y = 20x
Divide both sides by 8
Plug k back into the equation

18. Direct Variation Word Problems
The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.
K means 20 calories per ounce
Plug in 15 for x to solve the problem
Multiply to solve
y = 300
y = 20x
y = 20(15)

19. Direct Variation Word Problems
The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.
There are 300 calories in a 15-ounce glass of milk
y = 300

20. Direct Variation Word Problems
Laura typed 176 words correctly in 4 minutes. Assuming a direct variation, how many words can she type in 30 minutes?
A refund you get varies directly as the number of cans you recycle. If you get a $3.75 refund for 75 cans, how much should you get for 500 cans?

21. Direct Variation Word Problems
The number of miles driven varies directly with the number of gallons of gas used. Erin drove 297 miles on 9 gallons of gas. How far would she be able to drive on 14 gallons of gas?
The kinetic energy of a car varies directly with the square of the velocity of the car. A car with a velocity of 9 meters per second has 33,100 joules of kinetic energy. About how much kinetic energy does the same car have when traveling at 12 meters per second?

22. Direct Variation Word Problems
The height of a kite from the ground varies directly with the cube of the wind speed. A kite flies 8 feet high when the wind speed is 10 miles per hour. What is the height of a kite when the wind speed is 25 miles per hour?