Lesson 2-4: Represent Proportional Relationships with Equations

Property

In a relationship between two quantities, the independent variable is the quantity that is changed or controlled (the cause).
The dependent variable is the quantity that is measured or observed as a result (the effect).

Examples

Property

Proportional relationships can be represented by an equation of the form $y = kx$ or $y = rx$. In this equation:

• $x$ is the independent variable (input).
• $y$ is the dependent variable (output).
• $r$ (or $k$) is the constant of proportionality (the unit rate).

This equation shows that the output is always a constant multiple of the input.

Examples

• A machine prints 80 pages in 5 minutes. The unit rate is $r = \frac{80}{5} = 16$ pages per minute. The equation is $p = 16m$, where $p$ is pages and $m$ is minutes.
• The cost for apples is 2.50 dollars per pound. If $C$ is the total cost and $p$ is the number of pounds, the equation is $C = 2.5p$.
• A graph of a proportional relationship passes through $(4, 32)$. The unit rate is $\frac{32}{4}=8$. The equation representing this graph is $y = 8x$.

Explanation

An equation is like a powerful calculator for a proportional relationship. Once you find the constant rate ($r$), you can plug in any amount for $x$ to instantly find its corresponding amount $y$, without having to fill out a huge table.

Property

To find an unknown value in a proportional relationship, substitute the known value into the equation $y = kx$ and solve for the remaining variable.

• If $x$ is known: Substitute the value for $x$ and multiply by $k$ to find $y$.
• If $y$ is known: Substitute the value for $y$ and divide both sides by $k$ to find $x$.