What kind of variation that involves both the direct and inverse variation?

This section defines what proportion, direct variation, inverse variation, and joint variation are and explains how to solve such equations.

Proportion

A proportion is an equation stating that two rational expressions are equal. Simple proportions can be solved by applying the cross products rule.

If

More involved proportions are solved as rational equations.

Example 1

Solve

Apply the cross products rule.

The check is left to you.

Example 2

Solve

Apply the cross products rule.

The check is left to you.

Example 3

Solve

However, x = 4 is an extraneous solution, because it makes the denominators of the original equation become zero. Checking to see if

Direct variation

The phrase “ y varies directly as x” or “ y is directly proportional to x” means that as x gets bigger, so does y, and as x gets smaller, so does y. That concept can be translated in two ways.

• for some constant k.

  The k is called the constant of proportionality. This translation is used when the constant is the desired result.

• This translation is used when the desired result is either an original or new value of x or y.

Example 4

If y varies directly as x, and y = 10 when x = 7, find the constant of proportionality.

The constant of proportionality is

.

Example 5

If y varies directly as x, and y = 10 when x = 7, find y when x = 12.

Apply the cross products rule.

Inverse variation

The phrase “ y varies inversely as x” or “ y is inversely proportional to x” means that as x gets bigger, y gets smaller, or vice versa. This concept is translated in two ways.

• yx = k for some constant k, called the constant of proportionality. Use this translation if the constant is desired.

• y 1 x 1 = y 2 x 2.

  Use this translation if a value of x or y is desired.

Example 6

If y varies inversely as x, and y = 4 when x = 3, find the constant of proportionality.

The constant is 12.

Example 7

If y varies inversely as x, and y = 9 when x = 2, find y when x = 3.

Joint variation

If one variable varies as the product of other variables, it is called joint variation. The phrase “ y varies jointly as x and z” is translated in two ways.

• if the constant is desired.

• if one of the variables is desired.

Example 8

If y varies jointly as x and z, and y = 10 when x = 4 and z = 5, find the constant of proportionality.

Example 9

If y varies jointly as x and z, and y = 12 when x = 2 and z = 3, find y when x = 7 and z = 4.

Occasionally, a problem involves both direct and inverse variations. Suppose that y varies directly as x and inversely as z. This involves three variables and can be translated in two ways:

• if the constant is desired.

Example 10

If y varies directly as x and inversely as z, and y = 5 when x = 2 and z = 4, find y when x = 3 and z = 6.

What kind of variation is being described inverse or direct?

Identifying Direct and Inverse Relationships direct variation: both variables are increasing or decreasing. Inverse variation: one variable increases while the other decreases.

Which is an example of direct and inverse variation?

Numerically: Direct Variation: Because k is positive, y increases as x increases. So as x increases by 1, y increases by 1.5. Inverse Variation: Because k is positive, y decreases as x increases.

What is the relationship between direct and inverse variation?

Direct variation means when one quantity changes, the other quantity also changes in direct proportion. Inverse variation is exactly opposite to this.

What do you call the type of variation which involves more than one variation?

Joint Variation: If more than two variables are related directly or one variable changes with the change product of two or more variables it is called as joint variation.