Unit 4 — Linear Equations, Functions, and Mathematical Modeling

Welcome to Grade 8 Mathematics!

Mathematics allows us to describe patterns, predict outcomes, and solve complex problems.

During Grade 8, you will build the mathematical foundation needed for high school Algebra.

You will learn to represent relationships using equations, graphs, tables, and functions while applying mathematics to real-world situations.

Variables and Relationships

A variable represents a quantity that can change.

Examples include:

• distance traveled
• temperature
• population
• time
• cost

Mathematics allows us to describe how one variable changes in relation to another.

Recognizing these relationships helps us understand many natural and human-made systems.

Linear Equations

A linear equation represents a relationship that graphs as a straight line.

Examples include:

• y = 2x + 5
• y = -3x + 8
• y = ½x − 4

Linear equations describe relationships where one variable changes at a constant rate.

Slope

The slope of a line describes its rate of change.

Slope tells us:

• how steep a line is
• whether it rises or falls
• how rapidly one quantity changes compared to another

Positive slope rises from left to right.

Negative slope falls from left to right.

A slope of zero represents a horizontal line.

An undefined slope represents a vertical line.

The Coordinate Plane

The coordinate plane helps us represent mathematical relationships visually.

Each point has an ordered pair:

(x, y)

The horizontal axis is the x-axis.

The vertical axis is the y-axis.

Graphs help reveal patterns that may not be obvious in tables alone.

Functions

A function assigns exactly one output to each input.

Examples include:

• converting temperatures
• calculating wages
• determining travel distance
• predicting population growth

Functions help describe consistent mathematical relationships.

Tables, Graphs, and Equations

The same relationship can often be represented in several ways:

• a table
• a graph
• an equation
• a written description

Strong mathematicians move easily among these different representations.

Solving Multi-Step Equations

Many equations require several operations.

Example:

4x − 7 = 29

Add 7 to both sides:

4x = 36

Divide by 4:

x = 9

Always check your solution by substituting it into the original equation.

Mathematical Modeling

A mathematical model uses mathematics to describe a real-world situation.

Models help solve problems involving:

• finance
• engineering
• science
• transportation
• medicine
• environmental studies
• business

Models simplify complex situations while preserving important relationships.

Activity

Complete the following:

• graph ten linear equations
• determine the slope of ten lines
• solve twenty multi-step equations
• identify functions from tables and graphs
• create five mathematical models describing real-world situations

Explain your reasoning for every solution.

Practice

Answer these questions.

What does a variable represent?

What is a linear equation?

What does slope describe?

What is a function?

Why are graphs useful?

What is a mathematical model?

Review

Today you learned:

• variables describe changing quantities
• linear equations represent constant rates of change
• slope measures how rapidly quantities change
• functions assign one output to each input
• graphs reveal mathematical relationships
• mathematical models solve practical problems

These concepts form the foundation of Algebra and many scientific and engineering applications.

Looking Ahead

Next we will explore systems of equations, geometry, transformations, and introductory concepts that prepare students for high school geometry and advanced algebra.