Learn on PengiReveal Math, AcceleratedUnit 2: Proportional Relationships

Lesson 2-4: Represent Proportional Relationships with Equations

In Lesson 2-4 of Unit 2 from Reveal Math, Accelerated, 7th grade students learn how to represent proportional relationships with equations using the form y = kx, where k is the constant of proportionality. Students practice identifying the constant of proportionality from real-world contexts, such as clay animation photo rates and fuel-to-air ratios, and use it to write equations that model the relationship between two quantities.

Section 1

Identifying Independent and Dependent Variables

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

Section 2

The Proportional Relationship Equation

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.

Section 3

Solving for Unknown Values Using Proportional Equations

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 $x$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Identifying Independent and Dependent Variables

Property

Examples

Section 2

The Proportional Relationship Equation

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

Section 3

Solving for Unknown Values Using Proportional Equations

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 $x$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Identifying Independent and Dependent Variables

Property

Examples

Section 2

The Proportional Relationship Equation

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

Section 3

Solving for Unknown Values Using Proportional Equations

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 $x$ .