Chapter 5

Solving for the Unknown

An introduction to algebra: equations, balance, and real word problems

Algebra lets you find a missing number: how many hours you need to work, how many weeks until you reach a savings goal, how much data fits your budget. This chapter treats an equation like a balance scale and builds up to real word problems.

5.1

Equations and the Balance Principle

Learning objectives

After this section you will be able to:

Tell the difference between an expression and an equation.
Solve one-step equations with inverse operations.
Check a solution by substitution.

Real-life situation

The mystery bag

On a balance scale, a closed bag of candy plus 7 loose pieces balances exactly 19 loose pieces. How many pieces are in the bag?

Figure 5.1 x + 7 balances 19

The concept

A variable, often x, stands for an unknown number. An expression, like x + 7, has no equals sign. An equation, like x + 7 = 19, says two expressions are equal.

Think of an equation as a balanced scale. Adding or subtracting the same number on both sides, or multiplying or dividing both sides by the same nonzero number, keeps it balanced. To get the variable alone, undo what is being done to it with the inverse operation: subtraction undoes addition, division undoes multiplication, and the reverse.

Always check by substituting your answer into the original equation.

Equations and the Balance Principle: solution map

1
Identify the question
x + 7 = 19
2
Calculation 1
Subtract 7 from both sides: x = 19 − 7.
3
Calculation 2
x = 12.
4
Calculation 3
Check: 12 + 7 = 19. ✓
5
Interpret and check
x = 12 pieces

Figure 5.1.V Equations and the Balance Principle: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

variable cost
A cost that rises or falls with the number of items made or sold.
fixed cost
A cost that stays the same no matter how many items are sold, like rent.
break-even point
The sales level where revenue exactly equals total cost, so there is no profit and no loss.

Key terms

Variable: A letter standing for an unknown number.
Constant: A number that does not change.
Expression: Numbers, variables, and operations with no equals sign.
Equation: A statement that two expressions are equal.
Inverse operations: Operations that undo each other (+ and −, × and ÷).

Formulas & rules

Addition equation
x + a = b → x = b − a
Subtraction equation
x − a = b → x = b + a
Multiplication equation
a·x = b → x = b ÷ a
Division equation
x ÷ a = b → x = b × a

Worked examples

Example 1The mystery bag

x + 7 = 19

Solution

1Subtract 7 from both sides: x = 19 − 7.
2x = 12.
3Check: 12 + 7 = 19. ✓

Answer: x = 12 pieces

Example 2Multiplication

4n = 52

Solution

1Divide both sides by 4: n = 13.
2Check: 4 × 13 = 52. ✓

Answer: n = 13

Example 3Division

n ÷ 5 = 9

Solution

1Multiply both sides by 5: n = 45.
2Check: 45 ÷ 5 = 9. ✓

Answer: n = 45

Practice exercises 5.1

y − 14 = 30
6m = 84
k ÷ 3 = 12
25 + p = 61
0.5x = 8

Section 5.1 recap

An equation is a balance: whatever you do to one side, do to the other.
Isolate the unknown by undoing operations in reverse order.

Answer key with explanations: Section 5.1

Practice exercises

y = 44
m = 14
k = 36
p = 36
x = 16

Recap quiz

Division: Division undoes multiplication.
13: Add 9 to both sides.
9: Divide both sides by 3.

5.2

Multistep Equations

Learning objectives

After this section you will be able to:

Combine like terms.
Use the distributive property.
Solve equations with variables on both sides.

Real-life situation

Three bags and five extras

Three identical candy bags plus 5 loose pieces balance 26 loose pieces. Now there are two operations to undo.

Figure 5.2 3x + 5 balances 26

The concept

Undo operations in reverse order. In 3x + 5, the x was multiplied by 3 and then 5 was added, so subtract 5 first, then divide by 3.

Like terms have the same variable part, such as 5x and 2x. Combine them by adding or subtracting their coefficients.

The distributive property says a(b + c) = ab + ac. You can also divide both sides by the outside number first when it divides evenly.

When the variable is on both sides, subtract one variable term from both sides so it appears on only one side.

Multistep Equations: solution map

1
Identify the question
3x + 5 = 26
2
Calculation 1
Subtract 5: 3x = 21.
3
Calculation 2
Divide by 3: x = 7.
4
Calculation 3
Check: 3(7) + 5 = 26. ✓
5
Interpret and check
x = 7

Figure 5.2.V Multistep Equations: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

expense
Money a business spends to run, such as rent, supplies, or wages.
revenue
The total money a business brings in from sales, before any costs are paid.
budget
A plan that shows expected income and how it will be spent.

Key terms

Coefficient: The number multiplying a variable, like 3 in 3x.
Like terms: Terms with identical variable parts.
Distributive property: a(b + c) = ab + ac.

Formulas & rules

Two-step
ax + b = c → x = (c − b) ÷ a
Distribute
a(x + b) = ax + ab

Worked examples

Example 1Two steps

3x + 5 = 26

Solution

1Subtract 5: 3x = 21.
2Divide by 3: x = 7.
3Check: 3(7) + 5 = 26. ✓

Answer: x = 7

Example 2Parentheses

2(x + 4) = 30

Solution

1Divide by 2: x + 4 = 15.
2Subtract 4: x = 11.
3Check: 2(15) = 30. ✓

Answer: x = 11

Example 3Like terms

5x − 2x + 6 = 24

Solution

1Combine: 3x + 6 = 24.
2Subtract 6: 3x = 18.
3x = 6.

Answer: x = 6

Example 4Variables on both sides

7x + 3 = 4x + 15

Solution

1Subtract 4x: 3x + 3 = 15.
2Subtract 3: 3x = 12.
3x = 4.
4Check: 31 = 31. ✓

Answer: x = 4

Practice exercises 5.2

4a − 9 = 23
3(b − 2) = 21
8c + 2c = 150
6d + 5 = 2d + 29
x ÷ 4 + 3 = 10

Section 5.2 recap

Simplify each side first, then undo operations to isolate the unknown.
Check the solution by substituting it back into the original equation.

Answer key with explanations: Section 5.2

Practice exercises

a = 8
b = 9
c = 15
d = 6
x = 28

Recap quiz

Subtract 6: Undo the addition first.
5: x − 1 = 4, so x = 5.
7: 5x = 35.

5.3

Word Problems with the Unknown

Learning objectives

After this section you will be able to:

Translate words into algebraic expressions.
Use the blueprint to set up and solve an equation.
Check that the answer makes sense in the situation.

Real-life situation

The phone bill

Imani's prepaid plan charges a flat $40 per month plus $8 for each gigabyte of data. Last month's bill was $72. How many gigabytes did she use?

Keystone Mobile — October bill

Base plan ............ $40.00
Data (? GB × $8.00) ... $32.00
Total due ............ $72.00

Figure 5.3 Monthly bill summary (fictional carrier)

The concept

Translate key words. "Sum", "more than", "increased by" → add. "Less than", "decreased by" → subtract ("5 less than n" is n − 5). "Times", "twice", "of" → multiply. "Per", "split equally" → divide. "Is" → equals.

Blueprint for algebra: (1) Let a variable represent the unknown and write what it means. (2) Write the other quantities using that variable. (3) Write and solve the equation. (4) Check the answer in the words of the problem, not just the equation.

Goal: game console — $480.00

Already saved ........ $120.00
Weekly deposit ....... $30.00
Weeks needed ......... w = ?
Equation: 120 + 30w = 480

Figure 5.4 Savings goal tracker for a game console (hypothetical price)

Build the phone-plan equation

1
Base plan
$40
2
Data charge
$8 × unknown GB
3
Total bill
$72 → 40 + 8x = 72

Figure 5.3.V Build the phone-plan equation. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

monthly payment
The amount paid each month toward a loan or installment plan.
loan term
How long a loan lasts before it must be fully repaid.
total repayment cost
The full amount paid over a loan's life — principal plus all interest and fees.

Key terms

Translate: Rewrite a sentence as an algebraic expression or equation.
Consecutive whole numbers: Whole numbers in a row, written n, n + 1, n + 2.

Formulas & rules

Flat fee plus rate
Total = Fixed fee + Rate × Units
Parts of a total
Part₁ + Part₂ = Total

Worked examples

Example 1The phone bill

40 + 8g = 72

Solution

1Let g = gigabytes used.
2Subtract 40: 8g = 32.
3Divide by 8: g = 4.
4Check: $40 + 4 × $8 = $72. ✓

Answer: 4 GB

Example 2Saving for a game console

Darius wants a $480 game console (hypothetical price). He has $120 saved and puts $30 a week from his part-time job into savings. How many weeks until he can buy it?

Solution

1Let w = number of weeks.
2Saved so far + new savings = price: 120 + 30w = 480.
3Subtract 120: 30w = 360.
4Divide by 30: w = 12.
5Check: 120 + 30 × 12 = 120 + 360 = 480. ✓

Answer: 12 weeks

Example 3Splitting earnings

Tasha and Rob earned $540 together selling at a weekend market. Tasha earned twice as much as Rob. How much did each earn?

Solution

1Let r = Rob's earnings; Tasha = 2r.
2r + 2r = 540 → 3r = 540 → r = 180.
3Tasha = 2 × 180 = 360.
4Check: 180 + 360 = 540. ✓

Answer: Rob $180, Tasha $360

Example 4Gym classes

A gym charges $25 a month plus $10 per class. Ari's bill is $85. How many classes?

Solution

125 + 10c = 85.
210c = 60.
3c = 6.

Answer: 6 classes

Practice exercises 5.3

A number plus 18 is 47. Find the number.
You have $60 saved and add $15 a week. How many weeks until you have $240?
Three consecutive whole numbers add to 72. Find them.
Jordan has 3 times as many candies as Kim. Together they have 48. How many does each have?
A game bundle costs $360 (hypothetical). Kai has $90 and saves $45 a week. How many weeks?
A ride costs $3 to start plus $2 per mile. The fare was $19. How many miles?

Section 5.3 recap

Name the unknown with a variable, then translate the words into an equation.
A word-problem answer should be a sentence with units, checked against the question.

Answer key with explanations: Section 5.3

Practice exercises

n + 18 = 47 → 29
60 + 15w = 240 → w = 12 weeks
n + (n + 1) + (n + 2) = 72 → 3n + 3 = 72 → n = 23: 23, 24, 25
k + 3k = 48 → k = 12. Kim 12, Jordan 36
90 + 45w = 360 → w = 6 weeks
3 + 2m = 19 → m = 8 miles

Recap quiz

n − 5 = 12: "Less than" reverses the order: n − 5.
2n: Twice means multiply by 2.
Substitute it back into the situation: It must make the original statement true.

Chapter 5 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

5.1 Equations and the Balance Principle
Key formulas
- Addition equation: x + a = b → x = b − a
- Subtraction equation: x − a = b → x = b + a
- Multiplication equation: a·x = b → x = b ÷ a
- Division equation: x ÷ a = b → x = b × a
Key terms
Variable, Constant, Expression, Equation, Inverse operations
Remember
An equation is a balance: whatever you do to one side, do to the other.
5.2 Multistep Equations
Key formulas
- Two-step: ax + b = c → x = (c − b) ÷ a
- Distribute: a(x + b) = ax + ab
Key terms
Coefficient, Like terms, Distributive property
Remember
Simplify each side first, then undo operations to isolate the unknown.
5.3 Word Problems with the Unknown
Key formulas
- Flat fee plus rate: Total = Fixed fee + Rate × Units
- Parts of a total: Part₁ + Part₂ = Total
Key terms
Translate, Consecutive whole numbers
Remember
Name the unknown with a variable, then translate the words into an equation.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

Section 5.1
0.5x = 8
Show answer
x = 16
Section 5.2
x ÷ 4 + 3 = 10
Show answer
x = 28
Section 5.3
A ride costs $3 to start plus $2 per mile. The fare was $19. How many miles?
Show answer
3 + 2m = 19 → m = 8 miles
Review of Section 3.4
Write 8.5% as a decimal.
Show answer
0.085
Review of Section 4.3
Current balance $1,050.00, pending debits $220.45. Available balance?
Show answer
$829.55

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 4 Chapter 6