Direct Variation
Divect variation is a fancy name for a concept you are already familiar with. For example if you
are paid an hourly wage you know that the more hours you work the more you earn. Your earnings are related
(directly) to the hours
you work.
The relationship is this:
(money earned) = (hourly wage) • (hours worked).
The equation for this relationship looks like this:
E = W•H.
(E is your earnings, W is your hourly wage,
and H is the number of hours worked.)
This is a direct variation relationship. Your earnings are directly related to the hours
you work.
This relationship is also know as a direct proprotion because your earnings are directly proportional to the
hours that you work. (The ratio E/H is always equal to
W, that is what directly proportional means. For example if your wage is $7.50 per hour
then your earnings divided by the hours worked to earn them will always equal your wage which is $7.50)
All direct variation equations can be put in the same form as E = W•H.
Normally they are written as y = k•x or just y = kx.
K is the conatant of variation it
is the slope of the line y = kx.
K shows how y changes as x changes. If k
is positive (positive slope) then as x gets bigger y gets bigger,
if k is
negative (negative slope) then as x gets bigger y gets smaller.
Just like it is true that you earn zero dollars for zero hours of work, it is also true that if x
is zero in a direct variation equation y = zero.
All direct variation equations can be put in the form y = kx. Whenever
x is zero in a direct variation equation y will also be zero.
Equations of the form y = kx + b or
y = kx - b where b is a number other than zero
are not direct variation equations because when x is zero in any of these y
is not zero.
The table below shows some examples of equations that are
direct variation and some that are not direct variation:
| Direct Variation Equation | Not a Direct Variation Equation
|
|---|
| y = 3x | y = 3x +7
|
| y = -2x | y = -2x +6
|
| y = -(2/3)x | y = (4/5)x - 9
|
| y = (7/4)x | y = -(9/16)x + 5
|
In the table all the equations that are not direct variation are of the form y = mx + b.
The form y = mx + b is called the slope-intercept form of an equation because
the number in front of x, m
is called the slope of the line and b is the y-intercept of the line.
In the table all the equations that are not direct variation equations have values of b
that are not zero. The equations that have a b value of zero are all direct variation equations.
The key point to remember is that direct variation equations don't show a value for b
because it is always zero.
Three of the ways to state the relationship between your earnings and the hours worked are:
1. Your earnings vary directly
with
the hours worked.
2. Your earnings vary directly
as
the hours worked.
3. The hours worked and Your earnings
vary directly.
Each of these statements translates into the same equation: E = WH
For the first two statements the order of the variables in the equation is the same as in the statement, but notice
that in the third statement the order of the variables has to be switched to write the equation.
Although E and H can change in this equation,
your wage, W, never changes, it is constant. No matter how much you earn,
when you divide it by the hours worked you will always end up the same value W, your hourly wage.
Because W remains constant it is called the constant of variation.
Direct Variation - Summary of key points
-
All direct variation equations can be put in the form y = kx
-
If a series of x,y pairs (x,y)
are directly related, then the ratio y/x is the
same for all the pairs and is equal to k.
k = y/x
-
if y varies directly with x or if y varies directly as x
the equation must be written as:
y = kx. (Same order as in the problem)
-
if x and y vary directly then the equation must be written as:
y = kx. (Reverse order as in the problem)
- The constant of variation k is found by dividing y
by x:
k = y/x