Arithmetic and Pre-Algebra Foundations — MGU Mathematics Series
Part IV: Pre-Algebra Foundations
Variables · Expressions · Like Terms · Substitution · Evaluation
Chapter 9 of 14
MGU Mathematics Series  ·  Arithmetic and Pre-Algebra Foundations  ·  Library Textbook Edition

Chapter 9:
Variables
and Expressions

The formal bridge from arithmetic to algebraic thinking — variables as unknowns, changing quantities, and general patterns; writing, simplifying, and evaluating expressions

■ Part IV: Pre-Algebra Foundations ■

■ 13 Sections  •  Chapter 9 of 14 ■

Follows Chapters 1–8: Arithmetic, Fractions, Decimals, Ratios, Exponents, and Scientific Notation  ·  Prepares students for Chapter 10: Equations and Inequalities

Chapter 9 — Front Matter

Purpose and Overview

This chapter is the formal bridge from arithmetic into algebraic thinking. Every arithmetic operation you have practiced so far — whether adding fractions, simplifying decimals, applying percent change, or evaluating exponents — used specific numbers in specific situations. Algebra begins when you ask a different kind of question: what if the number is not yet known? What if the quantity changes depending on the situation? What if you want to write a rule that works for every number at once?

The answer to all three questions is the same tool: a variable.

Chapter 9 introduces variables as meaningful mathematical objects, not as mysterious symbols invented to make mathematics harder. It shows how letters represent unknown numbers, changing quantities, and general patterns. It explains how to build algebraic expressions from words and situations, how to name the parts of an expression, how to simplify expressions by combining like terms, and how to evaluate expressions by substituting specific values. It ends by showing how familiar arithmetic sentences can be rewritten as algebraic expressions, which is the first step toward writing and solving equations.

By the end of this chapter, you will see that algebra is not a new subject disconnected from arithmetic. It is arithmetic made general.

Opening Question

A plumber charges a flat fee of $75 for showing up, plus $40 for each hour worked. If you knew the plumber worked exactly three hours, you could compute the cost directly: 75 + 40 × 3 = 75 + 120 = $195. But what if you do not know yet how many hours the plumber will work? What if you want to write a single expression that gives the correct cost for any number of hours? The expression 75 + 40h does exactly that. When h = 3, it gives $195. When h = 5, it gives $275. One expression, written once, works for every possible number of hours. That is the power of a variable.

13Sections
10Objectives
16Key Terms
15+Worked Examples

Why This Chapter Matters

Algebra I depends entirely on the ideas in this chapter. Before you can solve equations, you must understand what an expression is. Before you can simplify equations, you must know how to combine like terms. Before you can graph a linear relationship, you must understand how a variable represents a changing quantity.

Beyond Algebra I, variables appear in science and engineering formulas such as d = rt and F = ma; in financial calculations such as interest formulas and budget models; in computer programming, where variables hold values that change as a program runs; in statistics, where variables represent measured quantities in data sets; and in every future MGU mathematics course beyond this textbook.

This chapter does not ask you to solve equations yet. It asks you to understand what expressions are, where they come from, how to read them, how to simplify them, and how to evaluate them. These are the prerequisites for everything that follows.

Learning Objectives

  1. 1Explain what a variable is and describe at least three ways variables are used in mathematics.
  2. 2Distinguish between variables used for unknown numbers, changing quantities, and general patterns.
  3. 3Translate verbal phrases into algebraic expressions and algebraic expressions into verbal phrases.
  4. 4Identify the terms, coefficients, constants, and factors of an algebraic expression.
  5. 5Recognize like terms and explain why only like terms can be combined.
  6. 6Simplify algebraic expressions by combining like terms.
  7. 7Evaluate an algebraic expression by substituting given values for variables.
  8. 8Apply substitution to evaluate formulas and real-world expressions.
  9. 9Recognize and correct common variable and expression mistakes.
  10. 10Rewrite familiar arithmetic sentences as algebraic expressions to prepare for equation writing.
Chapter 9 — Contents

Sections at a Glance

Sections 9.1 – 9.4
Understanding Variables: Three Roles, Three Purposes
What a Variable Is · Unknowns · Changing Quantities · Patterns
Section 9.1

What a Variable Is

In arithmetic, you work with specific numbers. You might add 12 and 7, multiply 3.5 by 4, or divide 18 by 6. The numbers are always given. The task is to compute. In algebra, some quantities are not given. They might be unknown, might change depending on the situation, or might stand for any number at all. Mathematicians represent these quantities with letters. A letter used to represent a number is called a variable.

The word "variable" comes from the Latin word for "changing," and that name fits well. A variable is a placeholder that can take on different values. It is not a fixed amount. It is a symbol that represents a number we do not yet know, a number that changes, or a number we want to keep general.

Any letter can be a variable. In algebra, the letters x, y, n, and t are especially common. In geometry formulas, l, w, and h are used for length, width, and height. In science, d, r, and t represent distance, rate, and time. Letters are chosen to be memorable, but any letter works mathematically.

Two Types of Mathematical Expressions

Arithmetic expression — contains only numbers and operation symbols, with no variables. Examples: 4 + 3, 12 × 5, 7² + 3.

Algebraic expression — contains at least one variable, possibly combined with numbers and operation symbols. Examples: x + 5, 3n, 2a − 7, 5x² + 3x − 9.

Neither type has an equal sign. An expression is a phrase, not a sentence. A sentence with an equal sign is called an equation.

Student Tip — What a Variable Is Not

A variable is not a label or a unit abbreviation. If a problem involves meters, you might choose m as a convenient letter, but m still represents a numerical value such as 3 or 7.5 or 100. In the expression 5m, the letter stands for a number. Think of a variable as an empty box that holds a number. The box has a label on the outside (the letter), and a number can be placed inside.

Example 9.1.1
Identifying Arithmetic Expressions, Algebraic Expressions, and Equations
  • ProblemState whether each of the following is an arithmetic expression, an algebraic expression, or neither: (a) 4 + 9   (b) 3x − 7   (c) n/5   (d) "The sum of a number and twelve"   (e) 6 = 6
  • ThinkAsk: does it contain a variable? Does it have an equal sign? Is it a phrase (expression) or a sentence (equation)?
  • Solve (a) 4 + 9 → Contains only numbers. Arithmetic expression. (b) 3x − 7 → Contains variable x. Algebraic expression. (c) n/5 → Contains variable n. Algebraic expression. (d) Verbal → A phrase in words; translates to n + 12. (e) 6 = 6 → Has an equal sign. This is an equation, not an expression.
Answer: (a) Arithmetic expression. (b) Algebraic expression. (c) Algebraic expression. (d) Verbal phrase → n + 12. (e) Equation.
Practice 9.1Concept Check and Skill Practice
  1. In your own words, explain the difference between an arithmetic expression and an algebraic expression.
  2. A student says that in the expression 4m, the letter m means "meters." Is this correct? Explain.
  3. State whether each is an arithmetic expression, an algebraic expression, or neither: (a) 14 + 3   (b) 7y + 2   (c) x = 10   (d) 15/3   (e) 2a + 5b − 1
  4. Write an algebraic expression for each description: (a) A number increased by 9   (b) Twice a number   (c) A number divided by 4   (d) Three less than a number
Section 9.2

Variables as Unknown Numbers

One of the oldest uses of a variable is to represent a number that is not yet known. Imagine a situation where someone is thinking of a number but has not told you what it is. You might call that hidden number n, or x, or any letter you choose, and then write mathematical statements about it. This is not magic. It is organized reasoning.

Consider this: you know that a number added to 8 gives 15. You can write n + 8 = 15. Here, n represents the unknown number. When a variable represents an unknown number, it has one specific value, even if you do not yet know what that value is. Your goal, when solving such a problem, is to find that specific value. This is what you will do in Chapter 10 when you solve equations.

Implied Multiplication

In algebra, when a number and a variable are written directly next to each other without any symbol between them, multiplication is implied. The expression 6x means 6 × x. The multiplication sign is dropped because it can be confused with the variable x. Always remember: a number directly beside a letter means multiply.

Example 9.2.1
Writing Expressions for Unknown Numbers
  • ProblemWrite an algebraic expression or equation for each: (a) A number decreased by 11. (b) Five more than a number equals 19. (c) A number multiplied by 6 equals 42. (d) A number divided by 3 — expression only.
  • ThinkIdentify the operation described, the known numbers, and place the variable where the unknown belongs.
  • Solve (a) "Decreased by" = subtraction. Expression: x − 11 (b) "Five more than" + "equals 19" Equation: x + 5 = 19 (c) "Multiplied by 6" + "equals 42" Equation: 6x = 42 (d) "Divided by 3" — expression. Expression: x/3
  • CheckRead each answer back in words to confirm the translation matches the original phrase.
Answer: (a) x − 11   (b) x + 5 = 19   (c) 6x = 42   (d) x/3
Practice 9.2Unknowns in Expressions and Equations
  1. When a variable represents an unknown number, does it have one value or many values? Explain.
  2. Write an algebraic expression for each phrase. Use n for the unknown: (a) A number plus 7   (b) A number minus 4   (c) Eight times a number   (d) A number divided by 5   (e) Three less than twice a number
  3. Write an equation for each situation: (a) A number added to 15 equals 28.   (b) Six times a number equals 54.   (c) A number divided by 4 equals 9.   (d) A number decreased by 7 equals 11.
Section 9.3

Variables as Changing Quantities

The second major use of a variable is to represent a quantity that changes from one situation to the next. Unlike an unknown number, which has one specific hidden value, a changing quantity can take on many different values depending on the input or the circumstances.

Return to the plumber example. The expression 75 + 40h contains h, which is not a mystery. There is no single "correct" value for h. The value of h depends on how many hours the plumber actually works. If the job takes 2 hours, h = 2. If it takes 6 hours, h = 6. The variable h represents a changing input, and the expression describes how the total cost changes as h changes. This idea is the foundation of functions, which are central to Algebra I.

ExpressionVariableWhat ChangesWhat Stays Fixed
75 + 40hh = hours workedTotal cost changes with hBase fee $75, rate $40/hr
s²s = side lengthArea changes with sThe squaring relationship
60tt = hours traveledDistance changes with tSpeed of 60 mph
2.5nn = notebooks boughtTotal cost changes with nPrice $2.50 each
Example 9.3.1
Building and Evaluating a Changing-Quantity Expression
  • ProblemA parking garage charges $4.00 for the first hour and $2.50 for each additional hour. Write an expression for the total cost for h additional hours, then evaluate for h = 0, h = 2, and h = 5.
  • ThinkThe $4.00 is fixed. The $2.50 per additional hour depends on h. Total cost = fixed charge + variable charge.
  • Solve Total cost = 4 + 2.5h h = 0: 4 + 2.5(0) = 4 + 0 = $4.00 h = 2: 4 + 2.5(2) = 4 + 5 = $9.00 h = 5: 4 + 2.5(5) = 4 + 12.50 = $16.50
  • CheckZero additional hours means only the base $4.00. Each additional hour adds $2.50 exactly. The pattern is consistent and reasonable.
Expression: 4 + 2.5h. Values: $4.00, $9.00, $16.50.
Practice 9.3Changing Quantities
  1. Explain the difference between a variable representing an unknown number and one representing a changing quantity. Give one example of each.
  2. Write an expression for each situation and identify the variable: (a) A taxi charges a base fee of $3 plus $1.75 per mile.   (b) A store sells apples for $0.60 each.   (c) A savings account has $500 and grows by $25 each week.
  3. For the expression 3 + 1.75m from problem 2(a), find the total cost when m = 4, m = 10, and m = 0.
Section 9.4

Variables in Patterns

A third use of a variable is to describe a general pattern — a rule that holds for all numbers, or for every member of a sequence. This is perhaps the most powerful use of variables, because a single expression can describe infinitely many situations at once. Consider the even numbers: 2, 4, 6, 8, 10, and so on. Instead of listing them all, you can describe them with the expression 2n, where n is any positive integer. When n = 1, you get 2. When n = 100, you get 200. The variable n lets you talk about all even numbers at once.

Example 9.4.1
Writing a Pattern Rule with a Variable
  • ProblemThe table shows a number pattern. Write an expression using the variable n, where n is the position number.

    Position: 1, 2, 3, 4, 5   |   Value: 5, 10, 15, 20, 25
  • ThinkThe value increases by 5 each time n increases by 1. When n = 1, value = 5. When n = 2, value = 10. The value is always 5 times the position number.
  • SolveExpression: 5n
  • Checkn = 1: 5(1) = 5 ✓    n = 3: 5(3) = 15 ✓    n = 5: 5(5) = 25 ✓
Answer: 5n
Example 9.4.2
Extending a Pattern Using an Expression
  • ProblemUsing the expression 5n from Example 9.4.1, find the value at position 12 and position 50.
  • Solve Position 12: 5(12) = 60 Position 50: 5(50) = 250
At position 12, the value is 60. At position 50, the value is 250.
Tip — Testing a Pattern Rule

When you write a pattern rule with a variable, always test it using at least two or three values from the table. If the expression gives the correct output for each input you test, it is a good candidate for the pattern rule.

Practice 9.4Pattern Rules
  1. Why is writing a pattern rule with a variable more useful than listing the terms of a pattern one by one?
  2. For each table, write an expression using n:
    (a) n: 1, 2, 3, 4    Value: 3, 6, 9, 12
    (b) n: 1, 2, 3, 4    Value: 4, 5, 6, 7
    (c) n: 1, 2, 3, 4    Value: 7, 14, 21, 28
  3. Using your expression from 2(b), find the value at position 20 and position 100.
  4. Extension: The odd numbers are 1, 3, 5, 7, 9, and so on. Write an expression for the n-th odd number. (Hint: test your expression against the list.)
Sections 9.5 – 9.7
Writing Expressions: Words, Structure, and Like Terms
Words to Symbols · Terms, Coefficients, Constants · Like Terms
Section 9.5

Writing Expressions from Words

One of the most important skills in algebra is translating between English and mathematics. Words describe situations; expressions describe them mathematically. Learning to move fluently between these two forms is essential not only for algebra, but for solving word problems in science, finance, and everyday life.

OperationCommon Verbal Phrases
Additionsum, plus, added to, increased by, more than, total, combined, greater than
Subtractiondifference, minus, subtracted from, decreased by, less than, reduced by, fewer than, take away
Multiplicationproduct, times, multiplied by, of, twice, double, triple, groups of
Divisionquotient, divided by, per, for each, ratio of, split equally
Student Warning — Order Reversal in Subtraction

"Five less than a number" means n − 5, not 5 − n. "A number subtracted from 10" means 10 − n, not n − 10. The phrase "less than" reverses the expected order. You are taking 5 away from the number, starting with the number. Always re-read your translation after writing it.

Example 9.5.1
Translating Single and Multi-Operation Phrases
  • ProblemWrite an algebraic expression for each: (a) The sum of a number and 8. (b) Twelve less than a number. (c) Four times a number, decreased by 3. (d) The quotient of a number and 7. (e) Five more than three times a number.
  • ThinkIdentify the operations, identify the numbers, and place them in the correct order. Watch carefully for reversal phrases.
  • Solve (a) "Sum" = addition. n + 8 (b) "Less than" reverses order. n − 12 (c) "Four times" then subtract. 4n − 3 (d) "Quotient" = division. n/7 (e) "Three times" then add 5. 3n + 5
  • CheckRead each expression back in words to verify it matches the original phrase.
Answers: (a) n + 8   (b) n − 12   (c) 4n − 3   (d) n/7   (e) 3n + 5
Example 9.5.2
Translating Expressions into Words
  • ProblemWrite a verbal phrase for each: (a) x + 4   (b) 2x − 9   (c) x/6 + 1
  • Solve (a) x + 4 → A number increased by four. (b) 2x − 9 → Twice a number, decreased by nine. (c) x/6 + 1 → A number divided by six, increased by one.
Multiple correct translations exist for each expression, as long as they describe the same operations in the correct order.
MGU Library Connection
  • For a complete list of key word translations used across the MGU Mathematics Series, see Appendix I: Word Problem Strategy Guide at the back of this textbook.
  • This guide also appears in the Algebra I textbook and is referenced throughout Chapter 13: Word Problems and Problem Solving.
Practice 9.5Translation Practice

Use n for the unknown number unless otherwise indicated.

  1. A number increased by 17
  2. The product of 9 and a number
  3. A number divided by 8
  4. Six less than a number
  5. Three times a number, increased by 4
  6. The sum of twice a number and 5
  7. Ten decreased by a number
  8. The quotient of a number and 12, minus 3
  9. Twice the sum of a number and 6
  10. One less than four times a number

Write a verbal phrase for each expression.

  1. n + 3
  2. 5n
  3. n − 9
  4. n/4
  5. 6n + 2
Error Analysis — Practice 9.5
"Eight less than a number" → 8 − n

Identify the error and write the correct expression. Explain why the order matters.

Section 9.6

Terms, Coefficients, and Constants

An algebraic expression is built from smaller parts called terms. Understanding the parts of an expression is like understanding the structure of a sentence: it lets you read, interpret, and manipulate expressions correctly.

A term is a single number, a single variable, or a product of numbers and variables that is separated from other terms by addition or subtraction signs. The minus sign in front of a term belongs to that term. When you list the terms of an expression, include the sign as part of the term.

Four Key Parts of Any Expression

Term — a single quantity connected to others by + or −. In 5x² − 3x + 7, the terms are 5x², −3x, and 7.

Coefficient — the numerical factor of a variable term. In −3x, the coefficient is −3. In x, the coefficient is 1 (implied). In −x, the coefficient is −1.

Constant — a term with no variable. Its value never changes. In 5x² − 3x + 7, the constant is 7.

Factor — one of the quantities multiplied within a product. In 5x, the factors are 5 and x.

Example 9.6.1
Identifying Parts of an Expression
  • ProblemFor the expression 8x − 4y + x − 3, identify: (a) the number of terms, (b) each term and its coefficient, (c) any constant terms.
  • ThinkLook for the addition and subtraction signs that separate terms. Each sign belongs to the term that follows it.
  • Solve Terms: 8x, −4y, x, −3 (four terms total) Coefficients: 8x → coefficient is 8 −4y → coefficient is −4 x → coefficient is 1 (implied) −3 → constant term, no coefficient Constant term: −3
  • CheckReassemble: 8x + (−4y) + x + (−3) = 8x − 4y + x − 3. Matches the original. ✓
Four terms. Coefficients: 8, −4, 1 (for x), none for constant. Constant term: −3.
Student Warning — The Implied Coefficient of 1

Students often forget that a term with no written coefficient has an implied coefficient of 1. The term x means 1x. The term −x means −1x. This matters when combining like terms. For example: 5x − x = 5x − 1x = 4x. Students who forget the implied 1 sometimes write 5x − x = 5, which is wrong.

Practice 9.6Identifying Expression Structure

For each expression, list all terms, identify each coefficient, and identify any constant terms.

  1. 3x + 7
  2. 2a − 5b + 1
  3. 9n − n + 4
  4. x² + 3x − 8
  5. −4m + 6n − m + 2
  6. What is the coefficient of −y?
Section 9.7

Like Terms

Not all terms in an expression can be combined. You can only add or subtract terms that are like terms. Like terms are terms that have exactly the same variable part — the same variables raised to the same powers. The coefficients do not need to match. Only the variable parts matter.

Why can you only combine like terms? Think about what the terms represent. The term 3x means "3 groups of whatever x is." The term 7x means "7 groups of whatever x is." Combining them gives "10 groups of whatever x is," which is 10x. But 3x and 7x² represent fundamentally different things. You cannot combine them any more than you can add 3 apples and 7 tables. They are not the same kind of quantity.

Like TermsUnlike TermsReason Unlike
3x and 7x3x and 3x²Different exponents
5x² and −2x²5x and 5yDifferent variables
4xy and 9xy4xy and 4xDifferent variable parts
6 and −117 and 7xOne constant, one variable term
x and −xx² and x³Different powers
Tip — Think of Variable Parts as Units

Think of variable parts as units. Just as inches and square inches are different units (you cannot add 5 inches + 3 square inches), x and x² are different "units." You can add inches to inches and square inches to square inches, but not to each other.

Practice 9.7Identifying Like and Unlike Terms
  1. What feature of a term determines whether it is "like" another term?
  2. Are 4xy and 4x like terms? Explain why or why not.
  3. For each pair, state whether the terms are like or unlike. If unlike, explain why.
    (a) 6x and 2x     (b) 3y and 3y²     (c) −5 and 11     (d) 8n and 8m     (e) 2a² and −7a²     (f) x and −x
  4. From this list, identify all pairs of like terms: 5x², −3x, 7, 2x², 9x, −1
Sections 9.8 – 9.10
Simplifying and Evaluating Expressions
Combining Like Terms · Evaluating · Substitution in Formulas and Tables
Section 9.8

Combining Like Terms

Once you can identify like terms, you can simplify expressions by combining like terms. Combining like terms means adding or subtracting the coefficients of like terms while keeping the variable part unchanged. The rule comes from the distributive property working in reverse: 3x + 7x = (3 + 7)x = 10x. The variable is factored out and the coefficients are added.

Simplifying an expression means writing it in its most compact form by combining all like terms until no two remaining terms are like each other.

Procedure for Combining Like Terms

3x + 7x = 10x    Add coefficients; keep variable part.

9a² − 4a² = 5a²    Subtract coefficients; keep variable part.

5y − y = 4y    Note: y = 1y, so 5y − 1y = 4y.

Steps: (1) Identify like-term groups. (2) Add or subtract coefficients within each group. (3) Write the simplified expression. (4) Confirm no further combining is possible.

Example 9.8.1
Combining Like Terms — Single Variable
  • ProblemSimplify 7x + 3 − 2x + 8.
  • ThinkThe x-terms are 7x and −2x. The constants are 3 and 8. Group and combine each type separately.
  • Solve 7x + 3 − 2x + 8 = (7x − 2x) + (3 + 8) = 5x + 11
  • CheckLet x = 2. Original: 14 + 3 − 4 + 8 = 21. Simplified: 10 + 11 = 21. ✓
Answer: 5x + 11
Example 9.8.2
Combining Like Terms — Two Variables
  • ProblemSimplify 4a − 3b + 2a + 7b − 5.
  • ThinkGroup the a-terms, the b-terms, and the constants separately.
  • Solve a-terms: 4a + 2a = 6a b-terms: −3b + 7b = 4b Constants: −5 (only one; nothing to combine) Result: 6a + 4b − 5
  • CheckThree unlike term groups remain. No further simplification is possible.
Answer: 6a + 4b − 5
Example 9.8.3
An Expression That Is Already Simplified
  • ProblemSimplify 5x² + 3x − 7.
  • Think5x² has x². 3x has x to the first power. −7 is a constant. All three are unlike.
  • SolveNo terms can be combined. The expression is already in simplest form.
Answer: 5x² + 3x − 7 (already simplified)
Student Warning — Exponents Do Not Change When You Add

A common mistake is to add the exponents when combining like terms. For example: 3x + 5x ≠ 8x². When you add like terms, you add only the coefficients. The variable part stays exactly the same. The correct answer is 8x. Exponents change only under multiplication rules (Chapter 7), never under addition or subtraction.

Practice 9.8Combining Like Terms

Simplify each expression by combining like terms. If already simplified, write "already simplified."

  1. 3x + 5x
  2. 9n − 4n + 2
  3. 7y + 3 − y − 1
  4. 4a + 2b − a + 3b
  5. 6m² + 3m − 2m² + m
  6. 10x − 4 + x + 7 − 3x
  7. 5p + 2q − 3p + q − 1
  8. 8 − 3n + 4n − 8
Error Analysis — Practice 9.8
4x + 2x = 6x²

Identify the error and explain what went wrong. What is the correct answer?

Section 9.9

Evaluating Expressions

To evaluate an algebraic expression means to find its numerical value by replacing each variable with a specific number and then computing the result using order of operations. This is called substitution: you substitute (replace) each variable with a given value.

The process: (1) Write the expression. (2) Replace each variable with the given value, using parentheses around substituted values to avoid sign errors. (3) Simplify using order of operations — grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. (4) State the result.

Example 9.9.1
Evaluating a Simple Expression
  • ProblemEvaluate 3x + 7 when x = 4.
  • Solve 3x + 7 = 3(4) + 7 = 12 + 7 = 19
  • Check3 × 4 = 12; 12 + 7 = 19. Order of operations followed (multiplication before addition). ✓
Answer: 19
Example 9.9.2
Evaluating with a Negative Value
  • ProblemEvaluate 5x − 2 when x = −3.
  • ThinkReplace x with (−3). Always use parentheses to keep the negative sign attached and visible.
  • Solve 5x − 2 = 5(−3) − 2 = −15 − 2 = −17
  • Check5 × (−3) = −15; −15 − 2 = −17. Signs handled correctly. ✓
Answer: −17
Example 9.9.3
Evaluating with Exponents
  • ProblemEvaluate x² + 3x − 4 when x = 2.
  • ThinkReplace x with (2) everywhere. Apply order of operations: exponents before multiplication, multiplication before addition and subtraction.
  • Solve x² + 3x − 4 = (2)² + 3(2) − 4 = 4 + 6 − 4 = 6
  • Check2² = 4; 3(2) = 6; 4 + 6 − 4 = 6. ✓
Answer: 6
Example 9.9.4
Evaluating with Two Variables
  • ProblemEvaluate 2a + 3b − 1 when a = 5 and b = −2.
  • Solve 2a + 3b − 1 = 2(5) + 3(−2) − 1 = 10 + (−6) − 1 = 10 − 6 − 1 = 3
  • CheckEach multiplication was done first. Additions and subtractions then left to right. Result: 3. ✓
Answer: 3
Student Warning — Parentheses with Negative Substitution

When you substitute a negative value for a variable, always use parentheses around it. For example, if x = −4 and you need to compute x², write (−4)² = 16, not −4² = −16. Without parentheses, −4² is parsed as −(4²) = −16. With parentheses, (−4)² = (−4)(−4) = +16. These are different values, and omitting the parentheses is one of the most common errors in early algebra.

Practice 9.9Evaluating Expressions
  1. 4x + 1 when x = 3
  2. 7 − 2n when n = 3
  3. 3y² when y = 4
  4. 5a − 3b when a = 2, b = 1
  5. x² − x + 2 when x = 5
  6. −2t + 8 when t = −3
  7. 24/x + x when x = 4
  8. 3(m + 2) − 4 when m = 5
Error Analysis — Practice 9.9
Evaluate x² when x = −3: x² = −3² = −9

What error did the student make? What is the correct answer and why?

Section 9.10

Substitution

Substitution is the general name for replacing a variable with a value and computing the result. It applies not only to single expressions, but to formulas, geometric rules, patterns, and tables of values. When a formula gives a rule, substitution is how you apply that rule to a specific situation. Consider the formula for the area of a rectangle: A = lw. If l = 9 and w = 4, substitution gives A = (9)(4) = 36 square units.

Example 9.10.1
Substitution in the Distance Formula
  • ProblemThe distance formula is d = rt, where d = distance, r = rate (speed), t = time. Find the distance if r = 55 mph and t = 3 hours.
  • Solved = rt = (55)(3) = 165
  • Check55 miles each hour for 3 hours = 165 miles. Reasonable. ✓
Answer: 165 miles.
Example 9.10.2
Substitution in a Table of Values
  • ProblemComplete the table for the expression 3n + 2 using n = 0, 1, 2, 3, 4.
  • Solve n = 0: 3(0) + 2 = 2 n = 1: 3(1) + 2 = 5 n = 2: 3(2) + 2 = 8 n = 3: 3(3) + 2 = 11 n = 4: 3(4) + 2 = 14
  • CheckEach value increases by 3, which matches the coefficient 3 in front of n. The pattern is consistent. ✓
Values: 2, 5, 8, 11, 14.
MGU Library Connection — Looking Ahead to Chapter 12
  • The table of values in Example 9.10.2 connects directly to Chapter 12: Coordinate Plane and Patterns, where you will learn to plot these pairs as points on a coordinate plane.
  • The expression 3n + 2 produces a straight-line graph when its values are plotted. This is the foundation of linear functions in Algebra I.
Practice 9.10Substitution in Formulas and Tables
  1. How is substitution related to evaluating an expression? Are they the same process?
  2. Use the formula P = 2l + 2w to find the perimeter for: (a) l = 10, w = 4   (b) l = 7, w = 7   (c) l = 15, w = 3
  3. Complete the table for the expression 4x − 1 using x = 0, 1, 2, 3.
  4. Complete the table for the expression x² + 1 using x = 0, 1, 2, 3.
Sections 9.11 – 9.13
Real Situations, Common Mistakes, and Arithmetic Made General
Expressions in Context · Seven Common Errors · From Arithmetic to Algebra
Section 9.11

Expressions in Real Situations

Algebraic expressions are not classroom abstractions. They come from, and return to, real situations. Every formula, pricing rule, recipe scale, science equation, and financial model is an algebraic expression at its core. When you write an expression from a real situation, ask yourself three questions: What is changing or unknown? (That is your variable — define it clearly with its units.) What is fixed? (That is your constant or fixed coefficient.) What operation connects them? (That is your expression structure.) Always define your variable before writing the expression. A vague variable produces a vague answer.

Example 9.11.1
A Cell Phone Plan
  • ProblemA cell phone plan charges $30 per month for unlimited texting and $0.10 per minute for phone calls. Define a variable, write an expression, evaluate for 120 minutes, and interpret the result.
  • DefineLet m = the number of minutes of phone calls made in one month.
  • Solve Total cost = 30 + 0.10m When m = 120: 30 + 0.10(120) = 30 + 12 = $42
  • InterpretIn a month with 120 minutes of phone calls, the total cost of the plan would be $42.
  • Check$0.10 × 120 = $12 for calls. Plus $30 base fee. Total $42. Reasonable. ✓
Expression: 30 + 0.10m.    Cost for 120 minutes: $42.
Example 9.11.2
A Geometry Situation
  • ProblemA fence is being built around a rectangular garden. The length is fixed at 18 feet. The width is not yet decided. Write an expression for the total fencing needed, then evaluate for a width of 7 feet.
  • DefineLet w = the width of the garden in feet. Perimeter formula: P = 2l + 2w, where l = 18.
  • Solve P = 2(18) + 2w = 36 + 2w When w = 7: P = 36 + 2(7) = 36 + 14 = 50 feet
Expression: 36 + 2w.    For width of 7 feet: 50 feet of fencing needed.
Practice 9.11Real-World Applications

For each problem: define your variable clearly, write an expression, and evaluate for the given values.

  1. A babysitter charges a flat fee of $10 plus $8 per hour. (a) Define a variable. (b) Write an expression for total earnings. (c) Evaluate for 3 hours and for 6 hours.
  2. A car travels at 65 mph. (a) Write an expression for total distance after t hours. (b) How far in 4 hours? In 2.5 hours?
  3. A baker uses 2.5 cups of flour per loaf of bread. (a) Write an expression for total flour for n loaves. (b) How much for 6 loaves? For 10 loaves?
  4. Write your own real-world expression. Describe the situation, define your variable, write the expression, choose two values for the variable, and interpret both results in full sentences.
Section 9.12

Common Mistakes with Variables

As students move from arithmetic into algebraic thinking, certain mistakes appear repeatedly. This section identifies them clearly, explains why they happen, and shows how to avoid them. Recognizing these mistakes in your own work, and in sample student work, is one of the fastest ways to improve your algebra skills.

Mistake 1 — Treating a Variable as a Label, Not a Number
"In 5m, the m stands for meters." Correct: m represents a numerical value. If m = 3, then 5m = 5(3) = 15. Variables are numbers, not unit abbreviations.
Mistake 2 — Forgetting the Implied Coefficient of 1
5x − x = 5 (treating x as having no coefficient) Correct: x = 1x, so 5x − x = 5x − 1x = 4x. The coefficient 1 is always present even when not written.
Mistake 3 — Adding Exponents When Combining Like Terms
3x + 5x = 8x² Correct: 3x + 5x = 8x. Add only the coefficients; the variable part stays the same. Exponents change under multiplication, not addition.
Mistake 4 — Combining Unlike Terms
4x + 3 = 7x Correct: 4x + 3 cannot be simplified. 4x and 3 are unlike terms. A variable term and a constant cannot be combined by addition.
Mistake 5 — Omitting Parentheses When Substituting a Negative Value
Evaluate x² at x = −4: x² = −4² = −16 Correct: (−4)² = (−4)(−4) = +16. Without parentheses, −4² is parsed as −(4²) = −16, which is a different value.
Mistake 6 — Reversing the Order in Subtraction Phrases
"Eight less than a number" → 8 − n Correct: n − 8. "Less than" reverses the expected order. You start with the number and subtract 8 from it.
Mistake 7 — Dropping a Negative Sign When Rearranging Terms
Rewriting 5 − 3x as 3x + 5 Correct: 5 − 3x = −3x + 5. The negative sign belongs to the term 3x. Moving a term without its sign changes the expression's value.
Practice 9.12Error Analysis

For each problem, identify the mistake and write the correct solution.

  1. 6x − x = 6
  2. 2x + 3x = 5x²
  3. 4a + 5 = 9a
  4. Evaluate x² at x = −2: answer given as −4
  5. "Seven less than a number" → 7 − n
  6. 3y − 7y = 4y
  7. 5 + 2x = 7x
Section 9.13

From Arithmetic Sentences to Algebraic Expressions

This final section brings together all the ideas from the chapter. It shows you how to look at familiar arithmetic sentences, patterns, and rules and rewrite them using variables. This is the last step before Chapter 10, where you will write and solve equations.

Arithmetic sentences are specific. Algebraic expressions are general. When you move from one to the other, you are moving from "this one case" to "all cases of this type." In each case, the algebraic expression captures the general rule. The specific number is replaced by a variable. The relationship stays the same.

Arithmetic (Specific)Algebra (General)
3 × 5 = 15 — one multiplication fact3n — any number multiplied by 3
Perimeter of a square with side 6: 4 × 6 = 24Perimeter of any square with side s: 4s
Earn $12/hr × 8 hrs = $96Earn $12/hr × h hours = 12h
Example 9.13.1
Generalizing a Savings Pattern
  • ProblemA student saves $15 every week. (a) Write arithmetic expressions for total savings after 1, 2, 3, and 4 weeks. (b) Identify the pattern. (c) Write a general algebraic expression for total savings after w weeks.
  • Solve (a) Week 1: 15 × 1 = 15 Week 2: 15 × 2 = 30 Week 3: 15 × 3 = 45 Week 4: 15 × 4 = 60 (b) The pattern multiplies 15 by the number of weeks. (c) Let w = the number of weeks. Total savings = 15w
  • CheckFor w = 3: 15(3) = 45. Matches the arithmetic computation. ✓
Answer: 15w
Example 9.13.2
Generalizing a Geometry Relationship
  • ProblemThe area of a triangle is half the base times the height. (a) Compute the area when base = 8 and height = 5. (b) Write the general algebraic formula using variables b and h. (c) Use the formula to find the area when b = 12 and h = 7.
  • Solve (a) Area = (1/2) × 8 × 5 = 20 square units (b) A = (1/2)bh (c) A = (1/2)(12)(7) = 84/2 = 42 square units
  • Check12 × 7 = 84; half of 84 = 42. Correct. ✓
Formula: A = (1/2)bh.    Area for b = 12, h = 7: 42 square units.
Practice 9.13Word-to-Symbol Problems
  1. A taxi driver earns a flat $5 tip plus $2 for each mile driven. (a) Write arithmetic expressions for trips of 3, 7, and 10 miles. (b) Write a general algebraic expression for any trip of m miles.
  2. A rectangle has a fixed length of 12 cm and a variable width. (a) Write arithmetic expressions for areas when the width is 3, 5, and 9 cm. (b) Write a general algebraic expression for the area when the width is w cm.
  3. Look at the following arithmetic sentences. Identify the general pattern and write an algebraic expression:
    5 + 5 × 2 = 15
    5 + 5 × 3 = 20
    5 + 5 × 4 = 25
    5 + 5 × n = ?
  4. The sum of a number and its double is always three times the number. (a) Verify this with the numbers 4, 7, and 10. (b) Write a general algebraic expression that shows why this is always true.
Chapter 9 — Back Matter

Chapter Summary

Key Terms Review

variable
A letter used to represent a number that is unknown, changing, or general. Variables are numbers, not labels.
algebraic expression
A mathematical phrase containing at least one variable, possibly combined with numbers and operation symbols, but no equal sign.
arithmetic expression
A mathematical phrase containing only numbers and operation symbols, with no variables.
term
A single number, a single variable, or a product of numbers and variables in an expression. Terms are separated by addition or subtraction signs.
coefficient
The numerical factor of a variable term. The coefficient of 5x is 5. The coefficient of x is 1 (implied). The coefficient of −x is −1.
constant
A term with no variable. Its value does not depend on any input and does not change.
like terms
Terms that have exactly the same variable part, including identical exponents. Only like terms may be combined by addition or subtraction.
unlike terms
Terms with different variable parts. They cannot be combined by addition or subtraction and must remain separate in a simplified expression.
combining like terms
Adding or subtracting the coefficients of like terms while keeping the variable part unchanged. The process by which expressions are simplified.
simplifying
Writing an expression in its most compact equivalent form by combining all like terms. A simplified expression has no two remaining terms that are like each other.
evaluate
To find the numerical value of an expression by substituting specific values for variables and applying order of operations.
substitution
Replacing a variable with a specific numerical value. Negative values must always be enclosed in parentheses before applying any operations.
factor
One of the quantities multiplied to form a product. In the term 5x, the factors are 5 and x.
verbal expression
A phrase in words that describes a mathematical relationship or operation. Verbal expressions can be translated into algebraic expressions.
pattern rule
An algebraic expression that describes the output for every position in a number sequence. Always verify a pattern rule by testing it against known values.
algebraic model
An expression or formula that uses a defined variable to represent a real-world rule. Always define variables with units before writing the model.

Concept Review Questions

  1. What is a variable? Why do mathematicians use letters to represent numbers?
  2. Describe three different roles a variable can play in an expression. Give a real-world example of each.
  3. What is the difference between an algebraic expression and an equation?
  4. Define term, coefficient, and constant. Give an example of each from the same expression.
  5. What does it mean for two terms to be "like terms"? What must be the same about them?
  6. Why can you not combine 3x and 4? What happens to the expression 3x + 4?
  7. What is the implied coefficient of the term −y?
  8. Why should you use parentheses when substituting a negative number?
  9. Describe at least three common mistakes students make when working with variables.
  10. Explain how an algebraic expression can represent a general pattern or rule.

Translation Practice

Write an algebraic expression for each phrase using n unless another variable is given. Then write a verbal phrase for each of the final expressions.

Part APhrases to Expressions
  1. A number increased by 14
  2. Twice a number, minus 9
  3. The quotient of a number and 5
  4. Three less than a number
  5. Four times the sum of a number and 2
  6. A number squared, plus 7
  7. Ten decreased by three times a number
  8. The product of 6 and a number, increased by 1
  9. Half of a number
  10. A number divided by 4, minus 3
  11. The sum of a number and its triple
  12. Nine more than the square of a number
Part BExpressions to Verbal Phrases
  1. n + 8
  2. 5n − 3
  3. n/6
  4. n² + 4
  5. 3(n + 1)
  6. 10 − 2n

Combining Like Terms Practice

Simplify each expression by combining like terms. If already simplified, write "already simplified."

  1. 4x + 9x
  2. 7y − 3y + 2
  3. 5a + 3b − 2a + b
  4. 6x² + x² − 4
  5. 8m − m + 3m
  6. 9 − 4n + n − 2
  7. 5x + 2y − x + 3y − 7
  8. 3p² + 5p − p² + p
  9. 4x² + 3x − 1
  10. 12 − 7 + 3x − x
  11. 2a + 2b + 2c
  12. 6r − 6r + 8

Evaluation Practice

Evaluate each expression for the given values. Show the parentheses substitution step.

  1. 5x + 3 when x = 4
  2. 2n − 7 when n = 6
  3. x² − 4 when x = 5
  4. 3a + 2b when a = 4, b = 1
  5. −2x + 10 when x = −3
  6. x/4 + 2 when x = 8
  7. n² + 3n − 5 when n = 2
  8. 4(x − 3) when x = 7
  9. 5m − 4n + 1 when m = 2, n = 3
  10. x³ when x = −2

Word Problems

For each problem: define your variable clearly, write an algebraic expression or model, and answer the specific question. Use the Read → Represent → Solve → Check → Explain method.

  1. A streaming service charges $9 per month for its basic plan and $0.99 for each additional movie rental. (a) Define a variable for the number of additional movies rented. (b) Write an expression for the total monthly cost. (c) Find the cost for 0, 3, and 7 additional movies.
  2. A garden nursery sells potted plants for $6 each with a flat delivery charge of $12. (a) Write an expression for the total cost of ordering p plants with delivery. (b) How much would 5 plants with delivery cost?
  3. A runner jogs at a steady speed of 8 minutes per mile. (a) Write an expression for the total time needed to run m miles. (b) How long would it take to run 6 miles? 13 miles?
  4. A bank account has a starting balance of $200. Every week the balance grows by $45. (a) Write an expression for the account balance after w weeks. (b) What will the balance be after 8 weeks? (c) What will the balance be after 0 weeks? Does this make sense?
  5. The perimeter of a regular hexagon is six times its side length. Write an algebraic expression for the perimeter with side length s, and find the perimeter when s = 7 cm.
  6. A business sells candles for $8 each. Materials cost $3 per candle and monthly overhead is $200. Write an expression for monthly profit on c candles sold. Find the profit for 60 candles.

Error Analysis

Each problem shows a student's work containing an error. Identify the error, explain what went wrong, and write the correct answer.

Problem 1
Simplify 5x + 2x: answer = 7x²

What is the error? What is the correct answer?

Problem 2
Simplify 8n − 3 + n: answer = 6n

What is the error? What is the correct answer?

Problem 3
"Six less than a number" → 6 − n

Is this correct? If not, write the correct expression and explain the error.

Problem 4
Evaluate x² + 1 when x = −4: answer = −16 + 1 = −15

What is the error? What is the correct answer?

Problem 5
Simplify 3a + 4b: answer = 7ab

Identify all errors. Write the correct form.

Problem 6
Simplify 9y − 9y: answer = 1

What went wrong? What is the correct answer?

Problem 7
Evaluate 3x − 2 when x = 5: 3(5) − 2 = 15 − 2 = 3

What arithmetic error was made? What is the correct answer?

Problem 8
Combine like terms in 6x² − 2x²: answer = 4

What is the error? What is the correct answer?

Chapter 9 Checkpoint

Chapter 9 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 9
Part A — Vocabulary and Concepts
  1. Define variable and give an example of each of its three roles.
  2. Explain the difference between a term and a coefficient. Give an example of each.
  3. What does it mean for two terms to be like terms? Give one example of a like pair and one of an unlike pair.
  4. Which of the following is an algebraic expression: 12 + 4, 3n − 7, x = 10? Explain your answer.
Part B — Translation
  1. Write an expression: "Four more than three times a number."
  2. Write an expression: "The square of a number, decreased by seven."
  3. Write an expression: "Twice the difference of a number and five."
  4. What is the coefficient of −y in the expression 4x − y + 3?
Part C — Simplifying
  1. Simplify: 8x − 3x + 5
  2. Simplify: 4a² − a + 2a² + 3a − 7
  3. Simplify: 7m + 4n − 3m + n − 2
  4. Simplify: 6r − 6r + 8
Part D — Evaluating
  1. Evaluate 5x − 2 when x = −3.
  2. Evaluate 2t² − 3t + 1 when t = −2.
  3. Evaluate 3m + 2n − 1 when m = 4 and n = −5.
  4. Evaluate x² + 3x − 4 when x = 2.
Part E — Applications
  1. A parking lot charges $2.50 per hour. Define a variable and write an expression for the total cost of parking for h hours. Find the cost for 3 hours and for 7 hours.
  2. The formula for the area of a trapezoid is A = (1/2)(b₁ + b₂)h. Find the area when b₁ = 6, b₂ = 10, and h = 4.
Part F — Error Analysis
  1. A student simplifies 5x + 4 − 2x + 3 and writes 7x + 7. Show the correct simplification step by step and identify the exact error.
  2. A student evaluates x² at x = −3 and writes −9. Identify and explain the error. What is the correct answer?
Bridge to Algebra I
Chapter 10: Equations and Inequalities

Chapter 9 ends at the edge of algebra. You have learned what variables are, how to write expressions from words and patterns, how to simplify expressions by combining like terms, and how to evaluate expressions through substitution. These are the vocabulary and mechanics of algebra.

In Chapter 10, you will build on everything here. An equation is a statement that two expressions are equal. Solving an equation means finding the specific value of a variable that makes the statement true. Every solving strategy in Algebra I begins with the skills you have practiced in this chapter. In Chapter 11, order of operations will be applied to more complex multi-step expressions. In Chapter 12, the substitution table from Section 9.10 will become a table of ordered pairs that you plot on a coordinate plane. The bridge has been built. Chapter 10 is on the other side.

MGU Library Connection
  • Glossary at the back of this textbook — complete variable and expression vocabulary
  • Appendix E: Formula Sheet — formulas introduced in Section 9.10 and 9.13
  • Appendix I: Word Problem Strategy Guide — complete signal word reference for Section 9.5
  • Chapter 10: Equations and Inequalities — applies all Chapter 9 expression skills to solving
  • Chapter 12: Coordinate Plane and Patterns — plots the substitution tables from Section 9.10
  • Algebra I, Chapter 1: Algebraic Language and Structure — extends this chapter's foundation
  • MGU Library Mathematics Dictionary — entries for variable, expression, coefficient, substitution