TELL WHETHER THE EQUATION REPRESENTS DIRECT VARIATION

Problem 1 :

y = - 8x

Solution :

This equation represents a direct variation, because it is in the form y = kx. The constant of variation is -8.

Problem 2 :

y - 4 = 3x

Solution :

Solve the equation for y.

y = 3x + 4

This equation does not represents a direct variation, because it is not in the form y = kx.

Problem 3 :

3y - 7 = 10x

Solution :

3y - 7 = 10x

Solve the equation for y.

3y = 10x + 7

y = (10/3) x + (7/3)

This equation does not represents a direct variation, because it is not in the form y = kx.

Problem 4 :

2y - 5x = 0

Solution :

2y - 5x = 0

Solve the equation for y.

2y = 5x

y = (5/2) x

This equation represents a direct variation, because it is in the form y = kx. The constant of variation is 5/2.

Problem 5 :

5y = -4x

Solution :

5y = -4x

Solve the equation for y.

y = (-4/5) x

This equation represents a direct variation, because it is in the form y = kx. The constant of variation is -4/5.

Problem 6 :

6y = x

Solution :

6y = x

Solve the equation for y.

y = (1/6) x

This equation represents a direct variation, because it is in the form y = kx. The constant of variation is 1/6.

Problem 7 :

Tell whether x and y show direct variation. Explain your reasoning

Solution :

By writing the values of x and y as ordered pairs, we get

(0, -2) (1, 1) (2, 4) and (3, 7)

Since the line does not pass through the origin, it cannot be a direct variation.

Problem 8 :

The time t (in hours) that it takes a group of volunteers to build a playground varies inversely with the number n of volunteers. It takes a group of 10 volunteers 8 hours to build the playground.

• Make a table showing the time that it would take to build the playground when the number of volunteers is 15, 20, 25, and 30.
• What happens to the time it takes to build the playground as the number of volunteers increases?

Solution :

When t is the number of hours of working, n be the number of volunteers.

Since it comes under inverse variation,

t ∝ k/n

t = k/n

8 = k/10

k = 80

t = 80/n

When the number of volunteers is 15, 20, 25, and 30.

When n = 15, t = 80/15 ==> 5.333

= 5 hours + 0.333... x 60 minutes

5 hours and 20 minutes

When n = 20, t = 80/20

4 hours

When n = 25, t = 80/25

= 3 hours + 0.2 x 60 minutes

3 hours and 12 minutes

When n = 30, t = 80/30

= 2 hours + 0.6 x 60 minutes

2 hours and 40 minutes

As the number of volunteers increases, the time it takes to build the playground decreases.

Because the time decreases as the number of volunteers increases, the time for 5 volunteers to build the playground should be greater than 8 hours.

t = 80/5 = 16 hours

Problem 9 :

The height of television screen varies directly with its width x.

a) Find the height when the width is 48 inches

b)  Sketch the graph of the equation.

Solution :

a. Use the equation to find the height when x = 48 inches.

y = (9/16) (48)

= 27 

So, when the width is 48 inches, the height is 27 inches.

b. To sketch a graph, plot the point (48, 27). Then draw the line that passes through this point and the origin.

Problem 10 :

Describe and correct the error in telling whether x and y show direct variation.

Solution :

By observing the graph, it does not pass through origin, then it cannot be a direct variation.

Problem 11 :

The table shows the profit y for recycling x pounds of aluminum. Tell whether x and y show direct variation.

Solution :

Relationship between the variables x and y in Direct variation is 

y = kx

Since the constant of variation is the same, the table represents direct variation.

Problem 12 :

The variables x and y vary directly. Use the values to write an equation that relates x and y.

a) y = 4; x = 2

b) y = 25; x = 5

Solution :

a) y = kx

y = 4; x = 2

4 = k(2)

4 = 2k

k = 4/2

k = 2

Applying the value of k in y = kx, we get

y = 2x

b) y = 25; x = 5

25 = k(5)

25 = 5k

k = 25/5

k = 5

Applying the value of k in y = kx, we get

y = 5x