A variation is a relation between a set of values of one variable and a set of values of other variables.
Direct variation
In the equation y = mx + b, if m is a nonzero constant and b = 0, then you have the function y = mx (often written y = kx), which is called a direct variation. That is, you can say that y varies directly as x or y is directly proportional to x. In this function, m (or k) is called the constant of proportionality or the constant of variation. The graph of every direct variation passes through the origin.
Example 1
Graph y = 2 x.
Example 2
If y varies directly as x, find the constant of variation when y is 2 and x is 4.
Because this is a direct variation,
y = kx (or y = mx)
Now, replacing y with 2 and x with 4, 
The constant of variation is 
.
Example 3
If y varies directly as x and the constant of variation is 2, find y when x is 6.
Since this is a direct variation, simply replace k with 2 and x with 6 in the following equation.
A direct variation can also be written as a proportion.
This proportion is read, “ y 1 is to x 1 as y 2 is to x 2.” x 1 and y 2 are called the means, and y 1 and x 2 are called the extremes. The product of the means is always equal to the product of the extremes. You can solve a proportion by simply multiplying the means and extremes and then solving as usual.
Example 4
r varies directly as p. If r is 3 when p is 7, find p when r is 9.
Method 1. Using proportions: Set up the direct variation proportion
Now, substitute in the values.
Multiply the means and extremes (cross multiplying) give
Method 2. Using y = kx:
Replace the y with p and the x with r.
p = kr
Use the first set of information and substitute 3 for r and 7 for p, then find k.
Rewrite the direct variation equation as 
.
Now use the second set of information that says r is 9, substitute this into the preceding equation, and solve for p.
Inverse variation (indirect variation)
A variation where 
is called an inverse variation (or indirect variation). That is, as x increases, y decreases. And as y increases, x decreases. You may see the equation xy = k representing an inverse variation, but this is simply a rearrangement of 
.
This function is also referred to as an inverse or indirect proportion. Again, m (or k) is called the constant of variation.
Example 5
If y varies indirectly as x, find the constant of variation when y is 2 and x is 4.
Since this is an indirect or inverse variation, 
Now, replacing y with 2 and x with 4, 
The constant of variation is 8.
Example 6
If y varies indirectly as x and the constant of variation is 2, find y when x is 6.
Since this is an indirect variation, simply replace k with 2 and x with 6 in the following equation.
As in direct variation, inverse variation also can be written as a proportion.
Notice that in the inverse proportion, the x 1 and the x 2 switched their positions from the direct variation proportion.
Example 7
If y varies indirectly as x and y = 4 when x = 9, find x when y = 3.
Method 1. Using proportions: Set up the indirect variation proportion.
Now, substitute in the values.
Multiply the means and extremes (cross‐multiplying) gives
Method 2. Using 
: Use the first set of information and substitute 4 for y and 9 for x, then find k.
Rewrite the direct variation equation as 
.
Now use the second set of information that says y is 3, substitute this into the preceding equation and solve for x.
