Algebra
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Algebra

Foundations: the rules of arithmetic, written in letters.

Algebra is not a new kind of math. It is the arithmetic you already know, with letters standing in for numbers so a single statement can describe every case at once. This book starts at the very bottom: what each operation means and the rules it obeys. Every later topic in algebra, and in engineering, is built on these rules.

Read straight through, or jump to any chapter from the contents. Each section introduces one idea, states it as a numbered rule, and works it through examples.

Rule A property that is always true.
Example The rule worked step by step.
Watch Out A common mistake.
Check Yourself Practice with hidden answers.
Chapter 1

The Language of Algebra

Before learning the rules, we need the words and symbols the rules are written in.

  1. 1.1 Numbers and the Number Line
  2. 1.2 Variables and Expressions
  3. 1.3 The Equals Sign

1.1Numbers and the Number Line

Algebra is arithmetic with a few new symbols. Everything in this book is built on ordinary numbers, so we start there.

The whole numbers are 0, 1, 2, 3, ... Extending them to the left of zero gives the integers: ..., −3, −2, −1, 0, 1, 2, 3, ... The numbers left of zero are negative, the numbers right of zero are positive, and zero itself is neither.

−5−4−3−2−1012345 negativepositive
Figure 1.1 · The integers on a number line. Opposites sit at equal distances on either side of zero.

Every number has an opposite: the number the same distance from zero on the other side. The opposite of 5 is −5, and the opposite of −5 is 5. The opposite of 0 is 0.

The distance a number sits from zero is its absolute value, written with vertical bars. Distance is never negative, so |5| = 5 and |−5| = 5.

Definition 1.1Absolute Value
|a| is the distance from a to 0

Absolute value strips the sign off a number and keeps its size. We will use it in Chapter 2 to add numbers with different signs.

Check Yourself
  1. What is the opposite of −12?
    Show answer
    12
  2. Evaluate |−7| and |7|.
    Show answer
    Both equal 7. They are the same distance from zero.
  3. Which is larger, −2 or −9?
    Show answer
    −2. It sits further right on the number line.

1.2Variables and Expressions

A variable is a letter that stands for a number. We use letters when the number is unknown, or when we want to say something true for every number at once.

An expression is a combination of numbers, variables, and operations, such as 3x + 5 or a − b. An expression has a value but makes no claim. It is like a phrase, not a sentence.

The parts of an expression separated by addition are its terms. In 3x + 5 the terms are 3x and 5. The number multiplying a variable is its coefficient: the coefficient of 3x is 3.

Note

Notation. Writing a number next to a variable means multiply. 3x means 3 · x, and ab means a · b. A raised dot · or parentheses 3(4) also mean multiplication. The symbol × is avoided because it looks like the letter x.

ExampleEvaluating an expression

Evaluate 3x + 5 when x = 4.

  1. Replace x with 4: 3(4) + 5
  2. Multiply: 12 + 5
  3. Add: 17
Check Yourself
  1. Name the terms of 7y + 2.
    Show answer
    7y and 2.
  2. What is the coefficient in 9n?
    Show answer
    9
  3. Evaluate 2a + 1 when a = 6.
    Show answer
    2(6) + 1 = 13

1.3The Equals Sign

An equation is a sentence that says two expressions have the same value: 3x + 5 = 17. The equals sign does not mean “here comes the answer.” It means the left side and the right side are the same number.

A good picture is a balance scale. Each side of the equals sign is a pan, and the equation says the pans are level. Every rule for solving equations (Chapter 8) is about keeping the scale level.

Note

Every rule in this book is written with variables, like a + b = b + a. That one line says the same thing about every pair of numbers: 2 + 3 = 3 + 2, −4 + 10 = 10 + (−4), and so on forever. That is the power of algebra: one sentence, infinitely many facts.

Check Yourself
  1. Is 4x − 1 an expression or an equation?
    Show answer
    An expression. There is no equals sign, so it makes no claim.
  2. Is 4x − 1 = 11 true when x = 3?
    Show answer
    Yes. 4(3) − 1 = 12 − 1 = 11.
Chapter 2

Rules for Addition

Addition is the first operation. Its rules say what you are allowed to rearrange.

  1. 2.1 What Addition Means
  2. 2.2 The Commutative Property
  3. 2.3 The Associative Property
  4. 2.4 The Additive Identity
  5. 2.5 The Additive Inverse
  6. 2.6 Adding Signed Numbers

2.1What Addition Means

Adding combines quantities. On the number line, adding a positive number moves you to the right, and adding a negative number moves you to the left.

ExampleAddition as movement
  • 3 + 4: start at 3, move 4 right, land on 7.
  • 3 + (−4): start at 3, move 4 left, land on −1.

2.2The Commutative Property

Rule 2.2Commutative Property of Addition
a + b = b + a

The order in which you add two numbers does not change the sum.

ExampleUsing the commutative property
  • 8 + 5 = 13 and 5 + 8 = 13.
  • x + 7 = 7 + x for every value of x.

“Commute” means to travel back and forth. The numbers can swap places.

2.3The Associative Property

Rule 2.3Associative Property of Addition
(a + b) + c = a + (b + c)

When adding three or more numbers, the grouping does not change the sum. You may add any neighboring pair first.

ExampleRegrouping to make arithmetic easier

Compute (17 + 6) + 4.

  1. Regroup: 17 + (6 + 4)
  2. 17 + 10 = 27
Note

Commutative is about order (who comes first). Associative is about grouping (where the parentheses go). Together they mean a list of additions can be rearranged and regrouped freely.

2.4The Additive Identity

Rule 2.4Additive Identity
a + 0 = a

Adding zero leaves any number unchanged. Zero is called the additive identity because the number keeps its identity.

ExampleIdentity
  • −9 + 0 = −9
  • 0 + y = y

2.5The Additive Inverse

Rule 2.5Additive Inverse
a + (−a) = 0

Every number plus its opposite equals zero. The opposite −a is called the additive inverse of a.

ExampleInverses cancel
  • 6 + (−6) = 0
  • −2.5 + 2.5 = 0
  • x + (−x) = 0

This rule is how equations get solved: to get rid of a number that is being added, add its opposite.

2.6Adding Signed Numbers

With the number line in mind, two procedures cover every case.

Rule 2.6aSame Signs
add the absolute values, keep the common sign

5 + 3 = 8 and (−5) + (−3) = −8.

Rule 2.6bDifferent Signs
subtract the smaller absolute value from the larger; take the sign of the larger

For −9 + 4: |−9| = 9 is larger, 9 − 4 = 5, and −9 is negative, so the sum is −5.

ExampleSigned sums
  • −7 + (−6) = −13 (same signs)
  • 12 + (−20) = −8 (different signs, 20 > 12, take the negative)
  • −3 + 10 = 7 (different signs, 10 > 3, take the positive)
Check Yourself
  1. −8 + (−5)
    Show answer
    −13
  2. −15 + 9
    Show answer
    −6
  3. 14 + (−14)
    Show answer
    0, by the additive inverse.
  4. Use regrouping to compute (38 + 19) + 1.
    Show answer
    38 + (19 + 1) = 38 + 20 = 58
Chapter 3

Rules for Subtraction

Subtraction has no new rules of its own. It is addition in disguise.

  1. 3.1 Subtraction Is Adding the Opposite
  2. 3.2 Subtracting a Negative
  3. 3.3 Order Matters
  4. 3.4 Grouping Matters
  5. 3.5 Subtraction with Zero

3.1Subtraction Is Adding the Opposite

Rule 3.1Definition of Subtraction
a − b = a + (−b)

To subtract a number, add its opposite. Every subtraction problem can be rewritten as an addition problem, and then the rules from Chapter 2 apply.

ExampleRewriting subtraction
  • 9 − 4 = 9 + (−4) = 5
  • 3 − 8 = 3 + (−8) = −5
  • −2 − 6 = −2 + (−6) = −8

3.2Subtracting a Negative

Rule 3.2Subtracting a Negative
a − (−b) = a + b

Subtracting a negative number is the same as adding the positive. The opposite of −b is b, so Rule 3.1 turns a − (−b) into a + b.

ExampleTwo negatives side by side
  • 5 − (−3) = 5 + 3 = 8
  • −10 − (−4) = −10 + 4 = −6
Watch Out

The two dashes in 5 − (−3) mean different things. The first is the operation subtract. The second is part of the number negative three. Always rewrite before computing.

3.3Order Matters

Rule 3.3Subtraction Is Not Commutative
a − b ≠ b − a (in general)

Swapping the numbers in a subtraction changes the answer. 7 − 2 = 5, but 2 − 7 = −5.

The two results are always opposites of each other:

a − b = −(b − a)

This is why rewriting as addition is so useful. Once 7 − 2 becomes 7 + (−2), it is commutative: 7 + (−2) = (−2) + 7. The negative sign travels with its number.

3.4Grouping Matters

Rule 3.4Subtraction Is Not Associative
(a − b) − c ≠ a − (b − c) (in general)

Moving the parentheses in a subtraction changes the answer.

ExampleTwo groupings, two answers
  • (10 − 4) − 3 = 6 − 3 = 3
  • 10 − (4 − 3) = 10 − 1 = 9

With no parentheses, subtract from left to right: 10 − 4 − 3 = 3. Chapter 7 covers this in detail.

3.5Subtraction with Zero

Rule 3.5Zero and Subtraction
a − 0 = a a − a = 0 0 − a = −a

Subtracting zero changes nothing. A number minus itself is zero. Zero minus a number is that number’s opposite.

Check Yourself
  1. Rewrite 6 − 11 as addition and compute.
    Show answer
    6 + (−11) = −5
  2. −4 − (−9)
    Show answer
    −4 + 9 = 5
  3. If 12 − x = 5, what is x − 12?
    Show answer
    −5, since x − 12 = −(12 − x).
  4. (20 − 8) − 2 and 20 − (8 − 2)
    Show answer
    10 and 14
Chapter 4

Rules for Multiplication

Multiplication shares addition's freedoms, plus one new rule for signs.

  1. 4.1 What Multiplication Means
  2. 4.2 The Commutative Property
  3. 4.3 The Associative Property
  4. 4.4 The Multiplicative Identity
  5. 4.5 The Zero Property
  6. 4.6 Multiplying Signed Numbers

4.1What Multiplication Means

For whole numbers, multiplication is repeated addition: 3 · 4 means three groups of four, 4 + 4 + 4 = 12. The numbers being multiplied are factors, and the result is the product.

Recall the notation from Section 1.2: 3 · 4, 3(4), and (3)(4) all mean the same thing, and 5x means 5 · x.

4.2The Commutative Property

Rule 4.2Commutative Property of Multiplication
ab = ba

The order of the factors does not change the product. Three groups of four and four groups of three are both 12.

ExampleReordering factors
  • 6 · 9 = 9 · 6 = 54
  • x · 5 = 5x (by convention, the number is written first)

4.3The Associative Property

Rule 4.3Associative Property of Multiplication
(ab)c = a(bc)

When multiplying three or more numbers, the grouping does not change the product.

ExampleRegrouping to simplify

4 · (25 · 7)

  1. Regroup: (4 · 25) · 7
  2. 100 · 7 = 700
ExampleSimplifying with variables

3(5x)

  1. Regroup: (3 · 5)x
  2. 15x

4.4The Multiplicative Identity

Rule 4.4Multiplicative Identity
a · 1 = a

Multiplying by one leaves any number unchanged. One is the multiplicative identity.

This is why a variable with no visible coefficient has coefficient 1: x = 1x.

4.5The Zero Property

Rule 4.5Multiplication by Zero
a · 0 = 0

Any number times zero is zero. Zero groups of anything is nothing.

Rule 4.5bZero Product Property
if ab = 0, then a = 0 or b = 0

The reverse is also true: the only way a product can be zero is if at least one factor is zero. This rule becomes essential when solving equations later in algebra.

4.6Multiplying Signed Numbers

Rule 4.6Signs of a Product
(+)(+) = + (−)(−) = + (+)(−) = −

Multiply the absolute values. If the signs are the same, the product is positive. If the signs are different, the product is negative.

ExampleSigned products
  • (−4)(−5) = 20
  • (−4)(5) = −20
  • (4)(−5) = −20

For longer products, count the negative factors. An even number of negatives gives a positive product; an odd number gives a negative product. So (−1)(−2)(−3) = −6, three negatives, odd.

Rule 4.6bMultiplying by Negative One
(−1)a = −a

Multiplying by −1 gives the opposite. This is why −x can be read as “negative one times x.”

Note

Why is a negative times a negative positive? It is forced by the other rules. Start from 0 = (−1) · 0 (Rule 4.5) and write 0 as 1 + (−1) (Rule 2.5):

0 = (−1)(1 + (−1)) = (−1)(1) + (−1)(−1) = −1 + (−1)(−1)

The only number that adds to −1 to make 0 is 1, so (−1)(−1) = 1. (The middle step uses the distributive property from Chapter 6.)

Check Yourself
  1. (−6)(7)
    Show answer
    −42
  2. (−3)(−3)(−3)
    Show answer
    −27 (three negatives, odd)
  3. Simplify 2(−4y).
    Show answer
    −8y
  4. If 7n = 0, what is n?
    Show answer
    0, by the zero product property.
Chapter 5

Rules for Division

Division is multiplication in disguise, with one firm restriction.

  1. 5.1 Division Is Multiplying by the Reciprocal
  2. 5.2 Dividing by One and by Itself
  3. 5.3 Zero in Division
  4. 5.4 Dividing Signed Numbers
  5. 5.5 Order and Grouping Matter

5.1Division Is Multiplying by the Reciprocal

The reciprocal of a nonzero number a is 1a. A number times its reciprocal is one:

Rule 5.1aMultiplicative Inverse
a · 1a = 1 (a ≠ 0)

The reciprocal is called the multiplicative inverse, the partner of the additive inverse from Rule 2.5.

Rule 5.1bDefinition of Division
a ÷ b = ab = a · 1b (b ≠ 0)

To divide by a number, multiply by its reciprocal. Just as subtraction became addition, division becomes multiplication.

ExampleRewriting division
  • 12 ÷ 3 = 12 · 13 = 4
  • 10 ÷ 12 = 10 · 2 = 20

5.2Dividing by One and by Itself

Rule 5.2One in Division
a1 = a aa = 1 (a ≠ 0)

Dividing by one changes nothing. Any nonzero number divided by itself is one.

5.3Zero in Division

Rule 5.3aZero Divided by a Number
0a = 0 (a ≠ 0)

Zero split into any number of groups leaves zero in each group.

Rule 5.3bDivision by Zero Is Undefined
a0 is undefined

No number can be the answer, so division by zero is not allowed.

Here is why. A division answer must check by multiplication: 123 = 4 because 4 · 3 = 12. Suppose 50 equaled some number c. Then c · 0 would have to equal 5. But Rule 4.5 says c · 0 = 0 for every c. No such c exists.

Watch Out

This is why so many rules carry the note “(a ≠ 0).” Whenever a variable ends up in a denominator, the rule only holds when that variable is not zero.

5.4Dividing Signed Numbers

Rule 5.4Signs of a Quotient
−a−b = ab −ab = a−b = −ab

The sign rules are the same as for multiplication: same signs give a positive result, different signs give a negative result.

ExampleSigned quotients
  • −24−6 = 4
  • −246 = −4
  • 24−6 = −4

5.5Order and Grouping Matter

Rule 5.5Division Is Not Commutative or Associative
a ÷ b ≠ b ÷ a (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Like subtraction, division depends on order and on grouping.

ExampleTwo groupings, two answers
  • (24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3
  • 24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12
Check Yourself
  1. −357
    Show answer
    −5
  2. What is the reciprocal of −23?
    Show answer
    −32, since (−23)(−32) = 1.
  3. 09 and 90
    Show answer
    0, and undefined.
  4. 8 ÷ 14
    Show answer
    8 · 4 = 32
Chapter 6

The Distributive Property

The one rule that connects multiplication to addition. Most of algebra's simplifying runs through it.

  1. 6.1 Distributing over Addition
  2. 6.2 Distributing over Subtraction
  3. 6.3 Distributing a Negative Sign
  4. 6.4 Factoring
  5. 6.5 Combining Like Terms

6.1Distributing over Addition

Rule 6.1Distributive Property
a(b + c) = ab + ac

Multiplying a sum is the same as multiplying each term and then adding. The factor outside is distributed to every term inside.

abac bc a b + c
Figure 6.1 · Area of the whole rectangle a(b + c) equals the sum of its parts ab + ac.

The picture shows why. A rectangle a tall and b + c wide has area a(b + c). Cutting it into two pieces gives areas ab and ac. Same rectangle, same area.

ExampleDistributing
  • 3(x + 4) = 3x + 3 · 4 = 3x + 12
  • 5(2y + 1) = 10y + 5
ExampleMental math
  • 7 · 102 = 7(100 + 2) = 700 + 14 = 714

6.2Distributing over Subtraction

Rule 6.2Distributing over Subtraction
a(b − c) = ab − ac

The same rule works for subtraction, because b − c is just b + (−c).

ExampleDistributing a negative factor

−2(x − 5)

  1. Multiply each term: (−2)(x) + (−2)(−5)
  2. −2x + 10
Watch Out

The most common mistake is distributing to only the first term: 3(x + 4) ≠ 3x + 4. The outside factor multiplies every term in the parentheses.

6.3Distributing a Negative Sign

Rule 6.3The Opposite of a Sum
−(a + b) = −a − b −(a − b) = −a + b

A negative sign in front of parentheses means multiply by −1 (Rule 4.6b). Distributing it flips the sign of every term inside.

ExampleRemoving parentheses
  • 10 − (x + 3) = 10 − x − 3 = 7 − x
  • −(4 − y) = −4 + y

6.4Factoring

Rule 6.4Factoring Out a Common Factor
ab + ac = a(b + c)

Reading the distributive property from right to left is called factoring. When every term shares a factor, pull it out front.

ExampleFactoring

Factor 6x + 9. Both terms share a factor of 3.

  1. 6x + 9 = 3 · 2x + 3 · 3 = 3(2x + 3)
  2. Check by distributing: 3(2x + 3) = 6x + 9 ✓

6.5Combining Like Terms

Like terms have exactly the same variable part: 3x and 5x are like terms; 3x and 5 are not.

Rule 6.5Combining Like Terms
bx + cx = (b + c)x

Add the coefficients and keep the variable. This is the distributive property (factoring) at work.

ExampleSimplifying an expression

4x + 7 + 2x − 3

  1. Rearrange (Rules 2.2, 2.3): 4x + 2x + 7 − 3
  2. Combine: 6x + 4
Check Yourself
  1. Expand 4(x + 6).
    Show answer
    4x + 24
  2. Expand −3(2a − 5).
    Show answer
    −6a + 15
  3. Factor 12y + 8.
    Show answer
    4(3y + 2)
  4. Simplify 9n − (2n + 4).
    Show answer
    9n − 2n − 4 = 7n − 4
Chapter 7

Order of Operations

When an expression mixes operations, everyone must agree on what to do first.

  1. 7.1 Why We Need an Order
  2. 7.2 The Order
  3. 7.3 Left to Right at the Same Level
  4. 7.4 Grouping Symbols and Fraction Bars
  5. 7.5 Negatives and Exponents

7.1Why We Need an Order

What is 2 + 3 · 4? Adding first gives 5 · 4 = 20. Multiplying first gives 2 + 12 = 14. Both can't be right, so mathematicians agreed on a convention. The agreed answer is 14.

7.2The Order

Rule 7.2Order of Operations
Grouping → Exponents → Multiply/Divide → Add/Subtract
  1. Grouping. Work inside parentheses, brackets, and other grouping symbols first.
  2. Exponents. An exponent is repeated multiplication: 32 = 3 · 3 = 9 and 23 = 2 · 2 · 2 = 8.
  3. Multiplication and division, from left to right.
  4. Addition and subtraction, from left to right.
ExampleFollowing the order

5 + 2(8 − 3)2

  1. Grouping: 5 + 2(5)2
  2. Exponent: 5 + 2(25)
  3. Multiply: 5 + 50
  4. Add: 55

7.3Left to Right at the Same Level

The memory aid PEMDAS can mislead. Multiplication does not come before division, and addition does not come before subtraction. Each pair is one level, worked left to right.

ExampleSame-level ties
  • 12 ÷ 3 · 2 = 4 · 2 = 8 (not 12 ÷ 6 = 2)
  • 10 − 4 + 3 = 6 + 3 = 9 (not 10 − 7 = 3)
Note

This connects to Chapters 3 and 5. Rewrite 10 − 4 + 3 as 10 + (−4) + 3 and the order no longer matters, because addition is commutative and associative.

7.4Grouping Symbols and Fraction Bars

When grouping symbols are nested, work from the innermost outward. A fraction bar is also a grouping symbol: simplify the entire top and the entire bottom before dividing.

ExampleNested grouping

2[3 + (6 − 4)]

  1. Innermost: 2[3 + 2]
  2. Next: 2[5] = 10
ExampleFraction bar
  • 8 + 45 − 2 = 123 = 4

7.5Negatives and Exponents

Watch Out

An exponent applies only to what it touches. In −32, the exponent touches only the 3, so square first and then take the opposite. In (−3)2, the parentheses put the negative inside.

−32 = −(3 · 3) = −9 (−3)2 = (−3)(−3) = 9
Check Yourself
  1. 4 + 6 ÷ 2
    Show answer
    4 + 3 = 7
  2. 18 ÷ 3 · 2
    Show answer
    6 · 2 = 12
  3. −22 + (−2)2
    Show answer
    −4 + 4 = 0
  4. 3(4) + 622 − 1
    Show answer
    183 = 6
Chapter 8

Rules of Equality

Every rule so far rewrites expressions. These rules let you solve equations.

  1. 8.1 Properties of Equality
  2. 8.2 Addition and Subtraction Properties
  3. 8.3 Multiplication and Division Properties
  4. 8.4 Putting the Rules to Work

8.1Properties of Equality

Rule 8.1Basic Properties of Equality
a = aif a = b, then b = aif a = b and b = c, then a = c

These are the reflexive, symmetric, and transitive properties. They say equality behaves the way you expect: a number equals itself, an equation can be read either direction, and things equal to the same thing are equal to each other.

Rule 8.1bSubstitution
if a = b, then a may replace b anywhere

This is what you did every time you evaluated an expression in Section 1.2.

8.2Addition and Subtraction Properties

Rule 8.2Adding or Subtracting the Same Amount
if a = b, then a + c = b + c and a − c = b − c

Adding or subtracting the same number on both sides keeps the scale level.

ExampleUndoing an addition

x + 7 = 12

  1. Subtract 7 from both sides: x + 7 − 7 = 12 − 7
  2. Additive inverse (Rule 2.5): x + 0 = 5
  3. Additive identity (Rule 2.4): x = 5

8.3Multiplication and Division Properties

Rule 8.3Multiplying or Dividing by the Same Amount
if a = b, then ac = bc and ac = bc (c ≠ 0)

Multiplying or dividing both sides by the same nonzero number keeps the scale level.

ExampleUndoing a multiplication

4x = 28

  1. Divide both sides by 4: 4x4 = 284
  2. Rule 5.2 (44 = 1) and Rule 4.4: x = 7

8.4Putting the Rules to Work

Solving an equation means using the rules to isolate the variable, one justified step at a time. Undo operations in the reverse of the order of operations: undo addition and subtraction first, then multiplication and division.

ExampleSolving a two-step equation

3x + 5 = 20

  1. Subtract 5 from both sides (Rule 8.2): 3x = 15
  2. Divide both sides by 3 (Rule 8.3): x = 5
  3. Check by substitution (Rule 8.1b): 3(5) + 5 = 15 + 5 = 20 ✓
ExampleUsing the distributive property first

2(x − 3) = 14

  1. Distribute (Rule 6.2): 2x − 6 = 14
  2. Add 6 to both sides (Rule 8.2): 2x = 20
  3. Divide both sides by 2 (Rule 8.3): x = 10
  4. Check: 2(10 − 3) = 2(7) = 14 ✓
Check Yourself
  1. Solve x − 9 = 4.
    Show answer
    x = 13
  2. Solve −5x = 30.
    Show answer
    x = −6
  3. Solve 4x − 7 = 21.
    Show answer
    4x = 28, so x = 7.
  4. Solve 3(x + 2) = 27.
    Show answer
    3x + 6 = 27, 3x = 21, x = 7.