A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation.

A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings.

The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

Tell whether each equation represents a direct variation. If so, identify the constant of variation.

**Example 1 : **

y = 4x

**Solution : **

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

**Example 2 : **

-2x + 3y = 0

**Solution : **

-2x + 3y = 0

Solve the equation for y.

Because -2x is added to 3y, add 2x to each side.

3y = 2x

Because y is multiplied by 3, divide each side by 3.

3y/3 = 2x/3

y = (2/3)x

This equation represents a direct variation, because it can be written in the form y = kx. The constant of variation is 2/3.

**Example 3 : **

3x + 2y = 6

**Solution : **

3x + 2y = 6

Solve the equation for y.

Because 3x is added to 2y, subtract 3x from each side.

2y = -3x + 6

Because y is multiplied by 2, divide each side by 2.

2y/2 = (-3x + 6)/2

y = -3x/2 + 6/2

y = -3x/2 + 3

This equation does not represent a direct variation, because it cannot be written in the form y = kx.

What happens if you solve y = kx for k?

y = kx

Divide each side by x (x ≠ 0).

y/x = kx/x

y/x = k

So, in a direct variation, the ratio y/x is equal to the constant of variation. Another way to identify a direct variation is to check whether y/x is the same for each ordered pair (except where x = 0)

Tell whether each relationship is a direct variation. Explain.

**Example 4 : **

**Solution : **

Write an equation that represents the relationship given in the table above.

y = 6x

Each y-value is 6 times the corresponding x-value.

This is a direct variation.

Because the equation y = 6x is in the form of y = kx, where k = 6.

**Example 5 : **

**Solution : **

Write an equation that represents the relationship given in the table above.

y = x - 4

Each y-value is 4 less than the corresponding x-value.

This is a not direct variation.

Because the equation y = x - 1 is not in the form of y = kx.

**Example 6 : **

The value of y varies directly with x, and y = 8 when x = 2. Find y when x = 5.

**Solution : **

It is given that y varies directly with x.

Write the equation for a direct variation.

y = kx

Substitute 8 for y and 2 for x.

8 = k(2)

8 = 2k

Divide each side by 2.

8/2 = 2k/2

4 = k

The equation is

y = 4x

Find y, when x = 5.

y = 4(5)

y = 20

**Example 7 : **

The three-toed sloth is an extremely slow animal. On the ground, it travels at a speed of about 6 feet per minute. Write a direct variation equation for the distance y a sloth will travel in x minutes. Then graph.

**Solution : **

Step 1 :

Write a direct variation equation.

Distance

is

6 feet per minute

times

number of minutes

y

=

6

**⋅**

x

Step 2 :

Choose values of x and generate ordered pairs.

Step 3 :

Graph the points and connect.

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