# Direct variation

A direct variation (D.V.) is a relationship between two variables x and y that can be written as

y = kx, k ≠ 0

This situation occurs when the ratio of two variables is constant

When y = kx, we say that y varies directly with x

When z = kt, we say that z varies directly with t

k is called the constant of variation

You are probably familiar with lighting. The distance that you are from lighting and the time it takes you to hear thunder could form a D.V.

Let's say you are 4 miles away from lighting and it takes you 20 seconds to hear thunder

And when you are 5 miles away, it takes you 25 seconds

The ratio of the time it takes to distance you are from lighting is constant since 20/4 = 25/5 = 5

The situation above represents a D.V. and you can find the equation that relates the time it takes to hear the thunder and the distance you are from lighting

Let's say that

x = distance from lighting

y = time it takes to hear the thunder

The time it takes to hear the thunder depends on the distance you are.

Thus, the time it takes to hear thunder varies directly with the distance you are

Then y = kx

Find k when

x = distance from lighting = 4 and

y = time it takes to hear thunder = 20

20 = k × 4

20/4 = k

5 = k

Therefore, y = 5x is the direct variation equation

Having that relationship is a good thing since you could now predict how long it will take to hear the thunder if you are 10 miles away

When x = 10, y = 5 × 10 = 50 seconds

Example #2:

A recipe for 6 cupcakes needs 1 cup of flour.The number of cupcakes you can make varies directly with the amount of flour

How many cupcakes can you make with 4 cups of flour?

Let x = amount of flour

y = number of cupcakes

Since y varies directly with x, y = kx

Find k when

x = 1 and

y = 6

6 = k × 1

6 = k

Therefore, y = 6x is the direct variation equation

When x = 4, y = 6 × 4 = 24. So you can make 24 cupcakes

Not every equation represents a D.V.

Examples of equations that are direct variations:

y = -4x

y = 5x

y = (-4/6)x

y = (2/3)x

y = 100x

Equations that are not

y = -4x + 2

y = 5x -5/7

y = (-4/6)x + 7

y = (2/3)x + 3

y = 100x + 5

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