Variation (Direct, Joint & Inverse)

Q: What is the difference between direct and inverse variation?

Direct ($y = kx$) means the variables rise together; inverse ($y = k/x$) means one rises as the other falls.

Q: What is joint variation?

Variation with more than one variable, like $y = kxz$, where $y$ varies directly with the product of several quantities.

Variation describes how one quantity depends on others through a constant $k$ . Direct variation is $y = kx$ ; inverse is $y = \dfrac{k}{x}$ ; joint variation combines several variables.

Every variation problem has the same two steps: use given values to find $k$ , then use $k$ to answer the question.

Direct and inverse

Direct variation, $y = kx$ , means $y$ grows in proportion to $x$ — double $x$ and $y$ doubles. Inverse variation, $y = \dfrac{k}{x}$ , means they move oppositely — double $x$ and $y$ halves.

Find $k$ by substituting a known pair, then rewrite the equation with that $k$ .

Solve in two steps

First, plug the given $x$ and $y$ into the correct model and solve for $k$ . That constant stays fixed for the whole relationship.

Then substitute the new $x$ (or $y$ ) with that $k$ to find the unknown value.

Worked examples

Example 1: direct variation

$y$ varies directly with $x$ , and $y = 15$ when $x = 5$ . Find $y$ when $x = 8$ .

Find k from y = kx

15 = k(5) \Rightarrow k = 3

Use k with the new x

y = 3(8)

Simplify

24

Answer: $y = 24$

Example 2: inverse variation setup

$y$ varies inversely with $x$ , and $y = 4$ when $x = 3$ . Find $k$ .

Use y = k/x

4 = \dfrac{k}{3}

Solve

k = 12

Answer: $k = 12$

Example 3: joint variation

$y$ varies jointly with $x$ and $z$ , and $y = 24$ when $x = 2$ and $z = 3$ . Find $y$ when $x = 5$ and $z = 4$ .

Find k from y = kxz

24 = k(2)(3) \Rightarrow k = 4

Use k with the new values

y = 4(5)(4)

Simplify

80

Answer: $y = 80$

Try one yourself

Common questions

What is the difference between direct and inverse variation?

Direct ( $y = kx$ ) means the variables rise together; inverse ( $\displaystyle y = \frac{k}{x}$ ) means one rises as the other falls.

How do I find k?

Substitute a known pair of values into the model and solve for the constant $k$ .

What is joint variation?

Variation with more than one variable, like $y = kxz$ , where $y$ varies directly with the product of several quantities.

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Variation (Direct, Joint & Inverse)

Direct and inverse

Solve in two steps

Worked examples

Example 1: direct variation

Example 2: inverse variation setup

Example 3: joint variation

Try one yourself

Common questions

Keep learning

Solving Rational Equations & Inequalities

Direct Variation

Graphing Rational Functions