Solving Direct Variation Equations - Krista King Math

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What are direct variation equations?

In this lesson we’ll look at solving direct variation relationships. Those are relationships that are of the form ???kx=y???, where ???k??? is a constant.

What is direct variation?

In a direct variation equation you have two variables, usually ???x??? and ???y???, and a constant value that is usually called ???k???.

The main idea in direct variation is that as one variable increases the other variable will also increase. That means if ???x??? increases ???y??? increases, and if ???y??? increases ???x??? increases. The number ???k??? is a constant so it’s always the same value throughout a direct variation problem.

The general form of a direct variation formula is ???y=kx???, where ???x??? and ???y??? are variables (numbers that change) and ???k??? is a constant (a number that stays the same).

In a direct variation problem, ???x??? and ???y??? are said to vary directly and ???k??? is called the constant of variation.

This lesson will help you find a missing term in a direct variation equation.

How to solve direct variation equations

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Solving for one variable when we know th value of the constant of variation and the other variable

Example

Two variables ???x??? and ???y??? vary directly. If the constant of variation, ???k???, equals ???20??? what is the value of ???y??? when ???x??? equals ???15????

Remember the general form of direct variation is ???y=kx???, and we know:

???k=20???

???x=15???

and we’re looking for ???y???.

So

???y=20(15)???

???y=300???

Let’s try another one.

Example

In a direct variation formula, the constant of variation, ???k???, has the quality ???5k=50???. If ???y=85??? what is the value of ???x????

Remember the general form of direct variation is ???y=kx???, and we know

???5k=50???

???\frac{5k}{5}=\frac{50}{5}???

???k=10???

and

???y=85???

So,

???85=10x???

Now let’s solve for ???x???.

???\frac{85}{10}=\frac{10x}{10}???

???8.5=x???

Sometimes it can be helpful just to think about how to solve two-step equations that are in the form of a direct variation problem.

Example

Solve the two-step equation.

If ???2k=14??? and ???kx=56???, what is the value of ???x????

We’ll solve the first equation for ???k???.

???2k=14???

???\frac{2k}{2}=\frac{14}{2}???

???k=7???

Now we’ll take the value we found for ???k??? and plug it into the second equation to solve for ???x???.

???kx=56???

???7x=56???

???\frac{7x}{7}=\frac{56}{7}???

???x=8???

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