Lesson 1: Inverse Variation

Property

For any two variables $x$ and $y$, $y$ varies directly with $x$ if

The constant $k$ is called the constant of variation. When two quantities are related by a proportion, we say they are proportional to each other.

To solve direct variation problems:

1. Write the formula for direct variation: $y = kx$.
2. Substitute the given values for the variables.
3. Solve for the constant of variation, $k$.
4. Write the equation that relates $x$ and $y$ using the value of $k$.

Examples

• If $y$ varies directly with $x$, and $y=45$ when $x=9$, find the equation. We use $y=kx$, so $45=k(9)$, which gives $k=5$. The equation is $y=5x$.
• The cost of juice, $C$, varies directly with the number of bottles, $n$. If 4 bottles cost 12 dollars, how much would 7 bottles cost? The relation is $C=kn$. We find $k$ from $12=k(4)$, so $k=3$. The equation is $C=3n$. For 7 bottles, the cost is $C=3(7)=21$ dollars.
• The distance, $d$, an ant crawls varies directly with time, $t$. If it crawls 120 cm in 3 minutes, how far can it crawl in 10 minutes? The formula is $d=kt$. Substituting gives $120=k(3)$, so $k=40$. The equation is $d=40t$. In 10 minutes, it crawls $d=40(10)=400$ cm.

Property

$y$ varies inversely with $x$ if

where $k$ is a positive constant. This relationship implies that the product of the variables is constant: $xy = k$.