Solving direct variation equations

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 $k$.

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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

$y=20(15)$

$y=300$

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Let’s try another one.

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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 $x$.

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

$8.5=x$

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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.

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The main idea in direct variation is that as one variable increases the other variable will also increase.

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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