Lesson 64: Identifying, Writing, and Graphing Inverse Variation

New Concept

Inverse variation is a relationship between two variables whose product is a constant. The equation is $xy = k$ or $y = \frac{k}{x}$.

What’s next

Next, you’ll use this definition to identify inverse variations, find missing values in a relationship, and graph the resulting curves.

Property

An Inverse variation is a relationship between two variables whose product is a constant. The equation $xy = k$ or $y = \frac{k}{x}$, where $k$ is a nonzero constant, defines an inverse variation between $x$ and $y$.

Explanation

Think of it like a seesaw! As one variable goes up, the other must go down to keep their product, $k$, perfectly balanced and constant. Unlike direct variation where they rise and fall together, in an inverse relationship, they move in opposite directions to ensure that the value of $xy$ always stays the same magical number.

Examples

• The relationship $xy = 15$ is an inverse variation because the product of $x$ and $y$ is a constant, 15.
• The relationship $y = 3x$ is a direct variation, not an inverse variation, because it's in the form $y=kx$.
• If 3 people take 8 hours to paint a fence, how long will it take 4 people? Here, the number of people and time vary inversely.

Isolating the variable $y$ is the key to determining if a relationship is an inverse variation. This example demonstrates one of the key ideas in this lesson, identifying an inverse variation.

Example Problem

Tell whether the relationship $3xy = 21$ is an inverse variation. Explain why.

Step-by-Step

1. Start with the given equation: $3xy = 21$.
2. To check if this is an inverse variation, we need to see if it can be written in the form $y = \frac{k}{x}$, where $k$ is a non-zero constant.
3. First, isolate the product $xy$ by dividing both sides by $3$: $xy = \frac{21}{3}$.
4. Simplify the right side: $xy = 7$.
5. Now, solve for $y$ by dividing both sides by $x$: $y = \frac{7}{x}$.
6. This equation matches the form $y = \frac{k}{x}$, with the constant of variation $k=7$. Therefore, the relationship is an inverse variation.

Property

The letter $k$ represents the constant of variation. In an inverse relationship, you can always find it using the formula $k=xy$.

Explanation

The constant of variation, $k$, is the secret code of the relationship! It's the one number that never changes, representing the fixed product of $x$ and $y$. If you can find $k$ by multiplying any known $(x, y)$ pair, you unlock the ability to write the master equation and solve for any other value in the relationship.

Examples

• If $y$ varies inversely with $x$, and $y=5$ when $x=4$, the constant of variation is $k = (4)(5) = 20$.
• To write an inverse variation equation where $y=2$ when $x=9$, first find $k$. $k = (9)(2)=18$. The equation is $y = \frac{18}{x}$.

Property

If $(x_1, y_1)$ and $(x_2, y_2)$ are solutions of an inverse variation, then their products are equal: $x_1y_1 = x_2y_2$.

Explanation

This is your ultimate shortcut! Since $x_1y_1$ and $x_2y_2$ both equal the same constant, $k$, they must equal each other. This lets you find a missing value in a new pair without first solving for $k$. Just set the product of the first pair equal to the product of the second and solve for the unknown value.

Examples

• If $y=5$ when $x=6$, find $x$ when $y=10$. Using the rule: $(6)(5) = x_2(10) \implies 30 = 10x_2 \implies x_2 = 3$.
• A teen earns 14 dollars an hour and works 10 hours. How many hours must they work to earn the same amount if their pay is 20 dollars an hour? $(14)(10) = (20)y_2 \implies 140 = 20y_2 \implies y_2 = 7$ hours.