Chapter 1 — Front Matter

Purpose and Overview

This chapter opens the study of algebra by explaining what algebra is, why it is useful, and how it grows naturally out of arithmetic. You have already learned to work with numbers directly — adding, subtracting, multiplying, dividing, and working through order of operations. Algebra extends that work by introducing symbols that stand in for numbers, patterns, and quantities that may be unknown or changing.

By the end of this chapter, you should be able to read and write algebraic expressions, evaluate them for given values, simplify them by combining like terms and applying the distributive property, and begin to see how algebra translates real situations into symbolic form. These skills will carry you through every topic in this book.

Opening Question

A cell phone plan charges a flat monthly fee of $25, plus $0.10 for every text message sent. If you send 80 messages in one month, how much do you owe? What if you send 200? What if you do not know how many messages you will send? By the end of this chapter, you will know how to write an expression that models this situation for any number of messages — and why a symbol is more useful than any one number alone.

14Sections

11Objectives

16Key Terms

10+Worked Examples

Learning Objectives

1Describe what algebra studies and explain how it extends arithmetic.
2Identify the role of variables as unknowns, inputs, and changing quantities.
3Name and distinguish constants, coefficients, terms, and factors in an expression.
4Translate verbal phrases into algebraic expressions.
5Evaluate algebraic expressions by substituting given values.
6Identify like terms within an expression.
7Combine like terms to simplify an expression.
8Apply the distributive property to expand or factor expressions.
9Recognize and produce equivalent expressions.
10Write algebraic models for simple real-world situations.
11Identify and avoid common mistakes with variables and expressions.

Chapter 1 — Contents

Sections at a Glance

1.1What Algebra StudiesStructure, relationships, and solving; why symbols matter; algebra as a language; connection to all later mathematics
1.2From Arithmetic to AlgebraArithmetic vs. algebraic statements; multiplication notation; division as fractions; testing with numbers
1.3Variables as Unknowns, Inputs, and Changing QuantitiesThree roles of variables; choosing variable names; defining variables with descriptions and units
1.4Constants, Coefficients, Terms, and FactorsDefining each part of an expression; monomials, binomials, trinomials; identifying from examples
1.5Writing Expressions from WordsSignal words for all four operations; order of subtraction and division; step-by-step translation method
1.6Evaluating ExpressionsSubstitution process; order of operations review; evaluating with negative values; real-world formulas
1.7SubstitutionWhy parentheses are essential; squaring negative values; common sign errors in substitution
1.8Like TermsDefining like and unlike terms; the variable part determines likeness; examples and counterexamples
1.9Combining Like TermsAdding and subtracting coefficients; why combining works; multi-term expressions; checking by substitution
1.10The Distributive Propertya(b + c) = ab + ac; distributing negatives; combining with like terms; checking examples
1.11Equivalent ExpressionsDefinition of equivalence; testing by substitution; three sources of equivalent forms
1.12Algebraic Models in Real SituationsWhat an algebraic model is; five-step modeling process; one- and two-variable models
1.13Common Mistakes with Variables and ExpressionsSix recurring errors; explanations and corrections; prevention strategies
1.14Preparing for Equations and FunctionsPreview of equations; preview of functions; readiness problems; bridge to Chapter 2

Section 1.1

What Algebra Studies

Arithmetic is the study of numbers and operations on numbers. Algebra extends arithmetic by asking a deeper question: what if we do not know the number, or what if the number can change? Algebra gives you a method for working in generality — writing 5x = 35 instead of computing one specific case at a time. The letter x holds the place of any number you are looking for.

What Algebra Studies	Description	Example
Structure	How expressions are built; why rules hold; how to rewrite in equivalent forms	`3(x + 2) = 3x + 6`
Relationships	How one quantity depends on another; formulas that work for any input	`P = 2l + 2w`
Solving	Finding values that satisfy an equation or inequality	`5x = 35 → x = 7`

Why Symbols?

In natural language: "Multiply a number by three, then add seven." In algebra: 3n + 7. The symbolic form is compact, precise, and ready to be manipulated. It also applies to any value of n, not only one case.

Practice 1.1Concept Check

In your own words, explain the difference between arithmetic and algebra. Give one example of each.
Why is it useful to write x instead of a specific number when describing a general situation?
A recipe calls for 3 cups of flour per batch. Explain why a symbol is more useful than listing specific cases.

Section 1.2

From Arithmetic to Algebra

Arithmetic Statement	Algebraic Statement
`6 + 6 + 6 = 18`	`3n = 18` when `n = 6`
`2 × 5 − 3 = 7`	`2x − 3 = 7`
`10 ÷ 2 = 5`	`x / 2 = 5`
`4² = 16`	`n² = 16`

Operations in Algebra

Multiplication is written without the × sign to avoid confusion with the variable x. Use a coefficient before a variable (3x), two variables side by side (ab), a number with parentheses (4(n + 2)), or a dot (3 · x). Division is written as a fraction: x/4 means x ÷ 4.

Tip — Test with Numbers

Whenever you are unsure whether an algebraic step is correct, substitute a specific number and check both sides. This habit of verifying with numbers is one of the most powerful tools in mathematics.

Practice 1.2Skill Practice

"A number multiplied by four equals twenty."
"Seven added to some value gives fifteen."
"A quantity divided by three gives eight."
"Nine less than a number equals two."
"A number squared equals twenty-five."

Section 1.3

Variables as Unknowns, Inputs, and Changing Quantities

Role	Description	Example
Unknown	A specific number you do not yet know. The goal is to find it.	`m + 12 = 20`; find m
Input	A value fed into a process; the output depends on it. Any value may be chosen.	`C = 2m + 3`; m is miles driven
Changing quantity	A quantity that varies over time or across situations.	`T = T₀ − 2h`; T drops as h grows

Tip — Naming Variables

Choose letters that suggest the quantity: t for time, d for distance, r for rate, h for height, C for cost. Always define your variable in words: "Let h = the height in meters."

Practice 1.3Concept and Skill

For each situation, identify the variable's role (unknown, input, or changing quantity) and explain your reasoning.

a. A plumber charges $50/hr. Let h = hours, C = total charge: C = 50h.
b. A student needs a score s where s ≥ 80 to pass. s is the student's unknown score.
c. A car travels 60 mph. Let t = time, d = distance: d = 60t.
Choose an appropriate variable name and define it clearly for: (a) gallons of water in a tank, (b) price of a single apple, (c) days since a deposit was made.

Section 1.4

Constants, Coefficients, Terms, and Factors

Four Key Parts of Any Expression

Constant — a fixed number with no variable. In 3x + 7, the constant is 7.

Term — a single quantity connected to others by + or −. In 5x² − 3x + 9, the terms are 5x², −3x, and 9.

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

Factor — one of the quantities multiplied in a product. In 5x², the factors are 5, x, and x.

Example 1.4.1

Identifying parts of an expression

ProblemIdentify the terms, coefficients, and constant in 6a² − 4a + 1.
ThinkSeparate the expression at every + and − sign. Each piece, including its sign, is one term.
Solve Terms: 6a², −4a, 1 Coefficient of 6a²: 6 Coefficient of −4a: −4 Constant term: 1
CheckReassemble: 6a² + (−4a) + 1 = 6a² − 4a + 1. ✓

Answer: Terms: 6a², −4a, 1. Coefficients: 6 and −4. Constant: 1.

Student Warning

When you see −x, the coefficient is −1, not 0 or just "negative." Writing −1 · x and writing −x mean exactly the same thing. Never drop the sign from the coefficient.

Practice 1.4Skill Practice

3x + 5
−2y² + 7y − 4
n
8
5m³ − m + 12
−x + 3

For each expression: list all terms, state the coefficient of each variable term, and identify any constant.

Section 1.5

Writing Expressions from Words

Operation	Signal Words
Addition	sum, total, more than, increased by, added to, plus, in all
Subtraction	difference, less than, decreased by, subtracted from, fewer than, minus, remaining
Multiplication	product, times, multiplied by, of, twice, triple, double
Division	quotient, divided by, per, ratio of, split equally

Student Warning — "Less Than"

"Five less than x" means x − 5, not 5 − x. The phrase "less than" reverses the expected order. You are taking 5 away from x, not the other way around. Always re-read the phrase after writing the expression.

Example 1.5.1

Translating a simple phrase

Problem"Eight more than a number n."
Think"More than" signals addition. Eight more than n means add 8 to n.
Solven + 8
CheckReading back: "n plus 8" or "8 more than n." Matches. ✓

Answer: n + 8

Example 1.5.2

Translating a multi-operation phrase

Problem"Three times a number, decreased by seven."
Think"Three times a number" → 3x. "Decreased by seven" → subtract 7.
Solve3x − 7

Answer: 3x − 7

Example 1.5.3

Translating "less than"

Problem"Twelve less than twice a number k."
Think"Twice a number" → 2k. "Twelve less than" means subtract 12 from 2k. The order matters.
Solve2k − 12
CheckIf k = 10: twelve less than 20 is 8. Expression: 2(10) − 12 = 8. ✓

Answer: 2k − 12

Practice 1.5Translation Practice

The sum of x and fourteen
A number y decreased by nine
Four times a number n
One-half of a number p
Six more than twice a number z
Ten less than a number m
Product of five and a number, divided by two
A number w squared, plus three
Three consecutive numbers starting at n
A number decreased by five, then multiplied by four

Section 1.6

Evaluating Expressions

Evaluating an expression means finding its numerical value when the variable is replaced by a specific number. The process: write the expression → replace each variable with the given value using parentheses → simplify using the order of operations (parentheses, exponents, multiplication and division left to right, addition and subtraction left to right).

Example 1.6.1

Evaluating a linear expression

ProblemEvaluate 4x − 3 when x = 5.
Solve 4x − 3 = 4(5) − 3 = 20 − 3 = 17
Check4 × 5 = 20; 20 − 3 = 17. ✓

Answer: 17

Example 1.6.2

Evaluating with a negative value

ProblemEvaluate 2x² + 3x − 1 when x = −2.
ThinkReplace every x with (−2). Use parentheses. Evaluate the exponent first.
Solve 2x² + 3x − 1 = 2(−2)² + 3(−2) − 1 = 2(4) + (−6) − 1 = 8 − 6 − 1 = 1
Check(−2)² = 4; 2(4) = 8; 3(−2) = −6; 8 − 6 − 1 = 1. ✓

Answer: 1

Example 1.6.3

Evaluating a real-world formula

ProblemA cell phone plan: total cost = 0.10t + 25 where t = messages. Find the cost for 80 and for 200 messages.
Solve t = 80: 0.10(80) + 25 = 8 + 25 = 33 t = 200: 0.10(200) + 25 = 20 + 25 = 45
CheckMore messages cost more. Base is $25; 80 msgs add $8; 200 msgs add $20. Reasonable. ✓

Answer: 80 messages → $33. 200 messages → $45.

Student Warning

Always write substituted values in parentheses. Writing 2(−2)² is clear and correct. Forgetting the parentheses leads to errors like treating −2² as −4 instead of +4, because −2² means −(2²) = −4, while (−2)² = +4.

Practice 1.6Evaluating Expressions

3x + 7 when x = 4
5n − 2 when n = 0
x² + 1 when x = 3
2a² − a + 4 when a = −1
6/x + 2 when x = 3
4t − t² when t = 4
3m + 2n when m = 5, n = −2
ab − b² when a = 3, b = 2

Section 1.7

Substitution

The most common substitution error is omitting parentheses when the substituted value is negative. Compare −3² with (−3)². The first means −(3²) = −9. The second means (−3)(−3) = +9. These are not the same. Whenever you replace a variable with a negative number, always enclose the entire value in parentheses before applying any exponents.

Example 1.7.1

Substitution with a negative value — squaring

ProblemEvaluate 3 − x² when x = −4.
ThinkReplace x with (−4). Evaluate the exponent before subtracting.
Solve 3 − x² = 3 − (−4)² = 3 − 16 = −13
Check(−4)² = 16; 3 − 16 = −13. ✓

Answer: −13

Common Substitution Error

3 − −4² = 3 − (−16) = 19

Error: −4² is computed as −(4²) = −16 rather than (−4)² = +16. Always use parentheses: replace x with (−4), not −4.

Practice 1.7Substitution Practice

Evaluate each expression. Show the parentheses step explicitly.

x² + 5 when x = −3
2 − n² when n = −5
4t² − 3t + 1 when t = −2
−x² + 2x when x = 4
a² − b² when a = 3, b = −3
(x + y)² when x = 2, y = −5

Section 1.8

Like Terms

Like terms are terms that have exactly the same variable part — the same variable or variables raised to the same powers. The coefficient does not affect whether terms are like. Only the variable part does.

Like Terms	Unlike Terms	Reason Unlike
`3x` and `7x`	`3x` and `3x²`	Different exponents
`5x²` and `−2x²`	`5x` and `5y`	Different variables
`4xy` and `9xy`	`4xy` and `4x`	Different variable parts
`6` and `−11`	`7` and `7x`	One constant, one not

Key Idea

Like terms have the same variable part. Unlike terms do not. Only like terms can be combined by addition or subtraction.

Practice 1.8Identifying Like Terms

In each group, identify which terms are like each other. State your reason.

4x, 7x, −2x², 5
3a², −a², 6a, a²
5mn, 3mn, −mn, 5m
8, −3, 8y, 15
6t³, t³, 4t², −t³

Section 1.9

Combining Like Terms

To combine like terms, add or subtract the coefficients and keep the variable part unchanged. This is valid because the distributive property works in reverse: 3x + 7x = (3 + 7)x = 10x. The variable is factored out and the coefficients are added.

Procedure

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.

Example 1.9.1

Combining like terms — single variable

ProblemSimplify 5x + 3 − 2x + 8.
Think5x and −2x are like terms. 3 and 8 are like terms (both constants). Group and combine.
Solve 5x + 3 − 2x + 8 = (5x − 2x) + (3 + 8) = 3x + 11
CheckLet x = 2. Original: 10 + 3 − 4 + 8 = 17. Simplified: 6 + 11 = 17. ✓

Answer: 3x + 11

Example 1.9.2

Combining like terms — multiple variable types

ProblemSimplify 4a² − 3a + 7 + 2a² + 5a − 1.
ThinkGroup by variable part: a²-terms, a-terms, constants.
Solve = (4a² + 2a²) + (−3a + 5a) + (7 − 1) = 6a² + 2a + 6
CheckLet a = 1. Original: 4 − 3 + 7 + 2 + 5 − 1 = 14. Simplified: 6 + 2 + 6 = 14. ✓

Answer: 6a² + 2a + 6

Student Warning

Do not combine unlike terms. 3x + 4x² cannot be simplified further — degree 1 and degree 2 are unlike. And exponents are not added by addition; only by multiplication. 3x + 4x² ≠ 7x³.

Practice 1.9Combining Like Terms

7x + 4x
9n − 3n + 5
6a − 2 + 3a + 7
4x² + x − 2x² + 5x
3m² − m + 4 − m² + 6m − 1
5 + 2y − 8 − y
4x + 3y − x + 2y
−6t² + 4t − 1 + t² − 4t + 8

Section 1.10

The Distributive Property

a(b + c) = ab + ac

a(b − c) = ab − ac

To multiply a factor by a sum or difference, multiply the factor by each term inside the parentheses, then add or subtract the results. This rule comes from the meaning of multiplication as repeated addition.

Example 1.10.1

Distributing a positive factor

ProblemSimplify 4(3x − 2).
Solve4(3x − 2) = 4 · 3x − 4 · 2 = 12x − 8
CheckLet x = 1: 4(3 − 2) = 4(1) = 4. Right side: 12(1) − 8 = 4. ✓

Answer: 12x − 8

Example 1.10.2

Distributing a negative factor

ProblemSimplify −3(2x − 7).
Solve −3(2x − 7) = −3 · 2x − (−3) · 7 = −6x + 21
CheckLet x = 2: −3(4 − 7) = −3(−3) = 9. Right: −12 + 21 = 9. ✓

Answer: −6x + 21

Example 1.10.3

Distributing then combining like terms

ProblemSimplify 2(3x + 4) − 5x.
ThinkDistribute first; then combine like terms.
Solve 2(3x + 4) − 5x = 6x + 8 − 5x = x + 8
CheckLet x = 3: original = 2(13) − 15 = 11. Simplified: 3 + 8 = 11. ✓

Answer: x + 8

Student Warning — Distributing a Negative

When you distribute a negative sign, every term inside the parentheses changes sign — not just the first. −(3x − 5) = −3x + 5, not −3x − 5. This is one of the most common errors in algebra.

Practice 1.10The Distributive Property

3(x + 5)
6(2n − 4)
−2(x + 3)
−4(3a − 1)
−(5x − 2)
7(m + n)
3(2x + 1) + 4x
5(n − 3) − 2(n + 1)
2(4a − 3) + 3(a + 5)
−3(x − 2) + 4(2x + 1)

Section 1.11

Equivalent Expressions

Two expressions are equivalent if they produce the same value for every possible value of the variable. Equivalent expressions are different symbolic forms of the same quantity. Every time you simplify, distribute, factor, or rearrange an expression, you are producing an equivalent form.

Source	Original Form	Equivalent Form
Distributing	`5(x + 2)`	`5x + 10`
Combining like terms	`3x + 4x − 1`	`7x − 1`
Factoring (reverse distribution)	`6x + 9`	`3(2x + 3)`

Example 1.11.1

Checking equivalence

ProblemAre 3(x − 4) and 3x − 12 equivalent?
ThinkApply the distributive property to the first expression and compare to the second.
Solve3(x − 4) = 3 · x − 3 · 4 = 3x − 12 ✓
CheckLet x = 5: first gives 3(1) = 3; second gives 15 − 12 = 3. ✓ Let x = 0: first gives −12; second gives −12. ✓

Answer: Yes. 3(x − 4) and 3x − 12 are equivalent expressions.

Practice 1.11Equivalent Expressions

Determine whether each pair is equivalent. Show your work.

2x + 6 and 2(x + 3)
4n − 4 and 4(n − 1)
3x + 5x and 15x
−(x − 4) and −x + 4
2(3x − 1) + 4 and 6x + 2
5(a + 2) − 3a and 2a + 10

Section 1.12

Algebraic Models in Real Situations

An algebraic model is an expression, equation, or formula that represents a real-world situation using symbols. Building a model: (1) read and identify all quantities; (2) define variables with descriptions and units; (3) identify the operations; (4) write the expression; (5) check by substituting a reasonable value.

Example 1.12.1

Building an expression model

ProblemA parking garage charges a flat entry fee of $4 plus $2 per hour. Write an expression for the total parking cost.
DefineLet h = the number of hours parked.
SolveTotal cost = 2h + 4
CheckFor 3 hours: 2(3) + 4 = $10. Three hours at $2/hr plus $4 base = $10. ✓

Answer: 2h + 4 dollars, where h = number of hours parked.

Example 1.12.2

A model with two variables

ProblemA farmer plants c corn rows (200 ears each) and s soybean rows (80 lbs each). Write expressions for total production.
ThinkTotal ears of corn: 200c. Total pounds of soybeans: 80s. Different units — cannot be combined.
Solve Corn: 200c ears Soybeans: 80s pounds

Answer: 200c ears of corn and 80s pounds of soybeans.

Practice 1.12Real-World Applications

Write an algebraic expression for each situation. Define your variables clearly.

A plumber charges $75 per hour. Write an expression for the total charge for any number of hours.
A store sells notebooks for $2.50 each. Write an expression for the cost of buying several notebooks.
A runner burns 8 calories per minute. Write an expression for total calories burned during a run.
A phone plan costs $30 per month plus $0.05 per minute of calls. Write an expression for the monthly bill.
A rectangle has a length 5 units more than its width w. Write expressions for the length and perimeter.
A student earns $9 per hour and received a $20 bonus this week. Write an expression for total weekly earnings.

Section 1.13

Common Mistakes with Variables and Expressions

Mistake 1 — Confusing 2x with x²

"Two times x and x squared are the same thing." Correct: 2x = x + x (repeated addition). x² = x · x (repeated multiplication). When x = 5: 2x = 10, but x² = 25.

Mistake 2 — Squaring without parentheses

When x = −3, x² = −9 Correct: (−3)² = (−3)(−3) = 9. Always substitute with parentheses: x² = (−3)² = 9.

Mistake 3 — Distributing to only the first term

3(x + 4) = 3x + 4 Correct: 3(x + 4) = 3x + 12. The factor 3 multiplies every term inside.

Mistake 4 — Distributing a negative sign incompletely

−(2x − 5) = −2x − 5 Correct: −(2x − 5) = −2x + 5. Distributing −1 changes every sign inside.

Mistake 5 — Combining unlike terms

3x + 4x² = 7x³

Correct: 3x + 4x² is already fully simplified. Different degrees cannot be combined; exponents are not added by addition.

Mistake 6 — Misreading "less than"

"Five less than x" written as 5 − x Correct: "Five less than x" means x − 5. You start with x and remove 5.

Practice 1.13Error Analysis

Each problem contains an error. Identify it and write the correct answer.

A student writes 5(x − 2) = 5x − 2. What is wrong? What is correct?
A student evaluates x² + 3 at x = −4 and gets −13. What went wrong?
A student says 4x + 3y = 7xy. Explain the error.
A student translates "six less than n" as 6 − n. Correct the translation.
A student distributes −(x − 8) as −x − 8. What is wrong? What is correct?

Section 1.14

Preparing for Equations and Functions

This chapter introduced the vocabulary and mechanics of algebraic expressions. The next step is equations — statements that two expressions are equal. An equation uses an equals sign to assert that two expressions have the same value. For example, 3x + 5 = 17 says there is a value of x that makes the left side equal to the right. Chapter 3 will teach you to find it. Every skill from this chapter — substitution, combining like terms, and the distributive property — will be used directly there.

Chapters 5 through 8 will introduce functions, which are rules that assign exactly one output to every input. The expressions you practiced here become function rules. The habits you built now — reading carefully, using parentheses, checking by substitution — are habits that serve every mathematician, scientist, engineer, and programmer.

Practice 1.14Readiness Check

Evaluate 2x + 5 for x = 6. Then find the value of x that makes the expression equal to 17.
Evaluate 3x − 4 for x = 3, x = 4, and x = 5. Which value makes the expression equal to 8?
Write an expression for "a number multiplied by 4, minus 3." What value makes this expression equal to 9?
A rental car costs $40/day plus a $15 fee. Write an expression for d days. For how many days can you rent the car for $175?
Two students simplify 3(x + 2) − x. Student A gets 2x + 6. Student B gets 2x + 2. Who is correct? Show all steps.

Chapter 1 — Back Matter

Chapter Summary

Sec 1.1Algebra extends arithmetic by using symbols for unknown, changing, or general quantities. It studies structure, relationships, and solving.
Sec 1.2Algebraic statements parallel arithmetic ones but use letters. Multiplication is written without ×; division is written as a fraction. Always test symbolic steps with numbers.
Sec 1.3Variables serve three roles: unknown (one value to find), input (any chosen value), and changing quantity (varies over time). Always define variables with descriptions and units.
Sec 1.4An expression is built from terms (connected by + and −). Each term has a coefficient and a variable part. Constants have no variable. Factors are quantities multiplied within a term.
Sec 1.5Verbal phrases translate to algebraic expressions using signal words for each operation. "Less than" and "subtracted from" reverse the order of subtraction. Always re-read after translating.
Sec 1.6Evaluating means substituting a specific value and simplifying with the order of operations. Negative values must be enclosed in parentheses to avoid sign errors with exponents.
Sec 1.7Parentheses are essential in substitution. −3² = −9 but (−3)² = 9. Always enclose negative substituted values in parentheses before applying any exponent.
Sec 1.8Like terms have exactly the same variable part, including the same exponents. Only like terms can be combined. The coefficient does not determine likeness; the variable part does.
Sec 1.9To combine like terms, add or subtract coefficients and keep the variable part unchanged. This is valid because the distributive property works in reverse. Unlike terms cannot be combined.
Sec 1.10The distributive property: a(b + c) = ab + ac. Multiply each term inside. When distributing a negative, every sign inside changes. Distribute before combining like terms.
Sec 1.11Two expressions are equivalent if they produce the same value for every variable value. Distribution, combining like terms, and factoring all produce equivalent expressions.
Sec 1.12An algebraic model uses a defined variable and an expression to represent a real-world rule. Always define variables, write the expression, and check by substituting a reasonable value.
Sec 1.13Six recurring errors: confusing 2x with x²; omitting parentheses on negative substitution; incomplete distribution; sign errors with negative factors; combining unlike terms; misreading "less than."
Sec 1.14Equations assert that two expressions are equal; solving finds the value that makes the statement true. Functions assign one output to each input. All Chapter 1 skills carry forward directly.

Key Terms Review

algebra

The branch of mathematics that uses symbols to represent unknown, changing, or general quantities and their relationships.

variable

A symbol (usually a letter) representing an unknown, an input, or a changing quantity.

constant

A fixed number in an expression that does not depend on any variable.

coefficient

The numerical factor multiplied by the variable part of a term. The coefficient of −3x is −3; of x it is 1.

term

A single quantity in an expression connected to others by addition or subtraction.

factor

One of the quantities multiplied together in a product.

expression

A mathematical phrase made of numbers, variables, and operations — no equals sign.

like terms

Terms with exactly the same variable part, including identical exponents. Only like terms may be combined.

unlike terms

Terms with different variable parts. They cannot be combined by addition or subtraction.

distributive property

The rule a(b + c) = ab + ac. Multiplication distributes over addition and subtraction.

equivalent expressions

Two expressions that produce the same value for every value of the variable.

evaluate

To find the numerical value of an expression by substituting a specific value for each variable.

substitution

Replacing a variable with a number or expression. Negative values must always be enclosed in parentheses.

simplify

To rewrite an expression in an equivalent but more compact form by distributing and combining like terms.

algebraic model

An expression, equation, or formula that represents a real-world situation using defined variables.

equation

A mathematical statement that two expressions are equal, written with an equals sign. Introduced fully in Chapter 3.

Skill Practice

Part AIdentifying Parts of Expressions

List all terms, the coefficient of each variable term, and any constants.

7x − 3
4a² + a − 9
12
−m + 6n − 2

Part BWriting Expressions from Words

The sum of a number and fifteen
Three times a number, minus eight
Twice the sum of a number and four
Seven less than four times a number
One-third of a number, increased by six
The square of a number, decreased by the number itself

Part CEvaluating Expressions

3x + 7 when x = −2
n² − 4n + 1 when n = 3
2a² + 3 when a = −1
(2x − 1) / 3 when x = 5
5t − t² when t = −2

Part DCombining Like Terms

9x + 4x − 3
5a² − 2a + a² + 6a
3m + 7 − m − 12
4x² + 3x − 2x² + x + 1
2p + 5q − p + 3q

Part EDistributive Property

4(3x + 1)
−2(5n − 3)
3(a + 4) + 2a
6(x − 2) − 3(x + 1)
−(4y − 7) + 2y

Word Problems

For each problem: define your variable, write an algebraic expression or model, and answer the specific question.

A gym membership costs $50 per month. Write an expression for total cost after any number of months. How much after one full year?
A store sells pens for $1.25 and notebooks for $3.00. Write an expression for the total cost of p pens and n notebooks. Find the total for 4 pens and 3 notebooks.
A car travels at 55 mph. Write an expression for the distance in t hours. How far in 3.5 hours?
A student scored 72 on the first test and p additional points on the second. Write an expression for the combined score. If the combined score is 155, how many points on the second test?
A field has length 3x meters and width (x + 5) meters. Write a simplified expression for the perimeter.
A business sells candles for $8 each. Materials cost $3 per candle; monthly overhead is $200. Write an expression for monthly profit on c candles. Find profit for 60 candles sold.

Error Analysis

Each problem shows a student's work. Identify the error and write the correct solution.

Problem 1

2(x + 3) + 4x = 2x + 3 + 4x = 6x + 3

What is the error? What is the correct answer?

Problem 2

Evaluate x² − 5 at x = −3: (−3)² − 5 = −9 − 5 = −14

What is the error? What is the correct answer?

Problem 3

3x² + 2x = 5x³

Identify all errors. Write the correct form.

Problem 4

"Nine less than twice a number n" → 9 − 2n

Is this correct? If not, write the correct translation and explain why.

Problem 5

−(x − 4) = −x − 4

Identify the error and write the correct answer.

Chapter 1 Checkpoint

Complete without assistance — assesses all key skills and concepts from Chapter 1

Section 1 — Vocabulary and Concepts

Define variable and give an example of each of its three roles.
Explain the difference between a term and a factor in the expression 5x².
What does it mean for two expressions to be equivalent?

Section 2 — Skills

Simplify: 8x − 3x + 5
Simplify: 4a² − a + 2a² + 3a − 7
Simplify: 3(2x − 5) + 4x
Simplify: −2(n + 4) − 3(n − 1)
Evaluate 5x − 2 when x = −3.
Evaluate 2t² − 3t + 1 when t = −2.
Evaluate 3m + 2n − 1 when m = 4 and n = −5.

Section 3 — Translation

Write an expression: "Four more than three times a number."
Write an expression: "The square of a number, decreased by seven."
Write an expression: "Twice the difference of a number and five."

Section 4 — Applications

A taxi charges $2.50 per mile plus a flat $3 base fee. Write an expression for the total fare for m miles. Evaluate for 8 miles.
A rectangle has width w and length 2w − 3. Write a simplified expression for the perimeter.

Section 5 — Error Analysis

A student simplifies 5(2x − 3) and gets 10x − 3. What is the error? What is the correct answer?
A student evaluates −x² + 4 at x = −2 and gets −8. Find and explain the error. What is correct?

Looking Ahead

Chapter 2: Real Numbers and Algebraic Properties

Chapter 1 gave you the vocabulary and mechanics of algebraic expressions. Chapter 2 will strengthen the number-system foundation on which all algebra is built — integers, rational numbers, irrational numbers, real numbers, absolute value, and operations with signed numbers. You will also see why the properties of arithmetic (commutative, associative, distributive, identity, inverse) are not arbitrary rules but the deep structural reasons that every algebraic manipulation is valid. Chapter 3 will then apply all of this to the central task of Algebra I: solving equations.

MGU Library Connection

Algebra Symbol Reference — Appendix D of this textbook
Pre-Algebra Readiness Review — Appendix A of this textbook
Arithmetic and Pre-Algebra Foundations, Chapters 8–10: Order of Operations, Signed Numbers, Properties of Operations
Algebra II, Chapter 1: Algebra II Readiness and Function Review
MGU Dictionary entries: expression, variable, coefficient, distributive property, equivalent expressions

Chapter 1:Algebraic Languageand Structure

Purpose and Overview

Learning Objectives

Sections at a Glance

What Algebra Studies

From Arithmetic to Algebra

Operations in Algebra

Variables as Unknowns, Inputs, and Changing Quantities

Constants, Coefficients, Terms, and Factors

Writing Expressions from Words

Evaluating Expressions

Substitution

Like Terms

Combining Like Terms

The Distributive Property

Equivalent Expressions

Algebraic Models in Real Situations

Common Mistakes with Variables and Expressions

Preparing for Equations and Functions

Chapter Summary

Key Terms Review

Skill Practice

Word Problems

Error Analysis

Chapter 1 Checkpoint

Chapter 1:
Algebraic Language
and Structure