Direct Variation

## Direct variation defined

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, that the ratio of the two values will always be the same. 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.

## Form of a direct variation equation

All (linear) direct variation equations can be put in the form  y = kx . When  x  is zero in a direct variation equation  y  will also be zero. That means that the y-intercept of all direct variation equations is always zero. Equations of the form  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.

Normally direct variation equations are written as  y = k•x  or just  y = kx.
K is the constant 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.

The table below shows some examples of equations that are
direct variation and some that are not direct variation:

Direct Variation EquationNot 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. The y-intercept of every direct variation equation is zero.

## Four ways to state the relationship between your earnings and the hours worked

1. The hours worked and Your earnings vary directly.

2. Your earnings vary directly as the hours worked.

3. Your earnings vary directly with the hours worked.

4. Your earnings are directly proportional to the hours worked.

Each of these statements translates into the same equation: E = WH
For the first statement the order of the variables in the equation is in the reverse order as in the statement, but notice that in the rest of the statements the order of the variables in the equations is the same as in the statement.

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 (linear) direct variation equations can be put in the form y = kx which is the same as y = mx + b but     b,  the y-intercept,   is always zero for direct variation equations.

• 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. For direct variation equations k is the rate of change or slope of the line.     k = y/x

• if x and y vary directly then the equation must be written as:
y = kx.    (Reverse order as in the problem)

• if  y varies directly as x
or y varies directly with x
or y is directly proportional to x
the equation must be written as:
y = kx.    (Same order as in the problem)

• The constant of variation k (the slope of the line) is found by dividing y by x:
k = y/x