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