Unit 7 — Algebraic Expressions, Equations, and Variables

Welcome back!

Algebra is often called the language of mathematics.

Instead of working only with known numbers, algebra allows us to describe patterns, represent unknown quantities, and solve problems that appear in science, engineering, economics, and everyday life.

Today you will strengthen your understanding of variables, expressions, equations, and mathematical reasoning.

Variables

A variable is a symbol that represents a number whose value may be unknown or may change.

Common variables include:

• x
• y
• n
• a
• b

Example:

x = 7

Variables allow mathematicians to describe general rules instead of individual examples.

Algebraic Expressions

An algebraic expression combines:

• numbers
• variables
• mathematical operations

Examples include:

• x + 5
• 3y
• 8 − n
• 4a + 7
• 2(x + 3)

Expressions do not contain an equals sign.

Expressions describe mathematical relationships.

Evaluating Expressions

To evaluate an expression, substitute a value for each variable.

Example:

If

x = 6

then

3x + 4

becomes

3(6) + 4

18 + 4 = 22

Always substitute carefully before performing calculations.

Equations

An equation states that two expressions are equal.

Examples:

x + 8 = 15

4y = 28

3a − 2 = 16

Equations ask us to determine the value of the unknown variable.

Solving One-Step Equations

Use inverse operations.

Example:

x + 9 = 21

Subtract 9 from both sides.

x = 12

Example:

5x = 35

Divide both sides by 5.

x = 7

Always check your answer by substituting it back into the original equation.

Two-Step Equations

Some equations require more than one step.

Example:

2x + 5 = 17

Step 1:

Subtract 5.

2x = 12

Step 2:

Divide by 2.

x = 6

Work carefully and complete one step at a time.

Inequalities

An inequality compares two quantities that are not necessarily equal.

Common symbols include:

• >
• <
• ≥
• ≤

Example:

x > 8

This means x can be any number greater than 8.

Inequalities often describe ranges of possible solutions.

Writing Equations from Word Problems

Many real-world situations can be written as equations.

Example:

A movie ticket costs $12.

Write an equation for the total cost of x tickets.

Answer:

12x = total cost

Mathematical models help us solve practical problems efficiently.

Activity

Complete the following:

• evaluate ten algebraic expressions
• solve ten one-step equations
• solve ten two-step equations
• graph five simple inequalities on a number line
• write five equations from real-world situations

Explain each step of your reasoning.

Practice

Answer these questions.

What is a variable?

How is an expression different from an equation?

What does it mean to evaluate an expression?

Why should you check your solution?

What is an inequality?

Why are variables useful in solving real-world problems?

Review

Today you learned:

• variables represent unknown or changing values
• expressions describe mathematical relationships
• equations can be solved using inverse operations
• inequalities describe ranges of values
• algebra helps model real-world situations
• checking solutions improves accuracy

Algebra provides a powerful language for describing patterns, solving problems, and understanding relationships throughout mathematics and science.

Looking Ahead

Next we will investigate geometry by studying angles, polygons, area, surface area, volume, and geometric reasoning, preparing for more advanced mathematics in Grades 7 and 8.