Lesson 2: Representing Proportional Relationships with Equations

Property

The variables $y$ and $x$ are proportional if

Examples

• The cost $C$ for gallons $g$ of gas is proportional. If 5 gallons cost 20 dollars, the constant is $k = \frac{20}{5} = 4$. The equation is $C = 4g$.
• The number of words $w$ you type is proportional to the minutes $m$ you spend typing. If you type 240 words in 4 minutes, the constant is $k = \frac{240}{4} = 60$. The equation is $w = 60m$.
• The length in centimeters $c$ is proportional to the length in inches $i$. Since 1 inch is 2.54 cm, the constant of proportionality is $k = 2.54$. The equation is $c = 2.54i$.

Explanation

Proportional variables have a constant ratio. This means one variable is always a fixed multiple of the other. Think of it like a recipe: doubling the ingredients doubles the serving size. Their graph is a straight line through the origin (0,0).

Property

If you know one pair of corresponding values $(x, y)$ from a proportional relationship (where $x \neq 0$), you can find the constant of proportionality $k$ by calculating $k = \frac{y}{x}$. The equation for the relationship is then $y = kx$.

Examples

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.