Algebra as Language Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Algebra as Language.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The perspective that algebra is a formal language with syntax and grammar for expressing mathematical ideas and relationships precisely.

Just as English has grammar, algebra has rules for combining symbols meaningfully.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Algebra-as-language sees expressions as sentences whose syntax rules decide what is meaningful.

Common stuck point: The procedure for algebra as language is the easy part; the trap is reading $5x$ as a concatenated number or 'five and x'. Asking "Am I treating these symbols as a language with grammar that must be read or written correctly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I treating these symbols as a language with grammar that must be read or written correctly?

Worked Examples

Example 1

easy

Translate into algebra: 'Five more than twice a number is seventeen.'

Answer

2x + 5 = 17

First step

Step 1: 'a number' →

x

Full solution

2
Step 2: 'twice a number' → $2x$ .
3
Step 3: 'Five more than' → $2x + 5$ .
4
Step 4: 'is seventeen' → $= 17$ . Equation: $2x + 5 = 17$ .

Algebra serves as a precise language for expressing relationships. Translating word problems into equations is the first step in solving them — the equation captures the structure of the problem.

Example 2

medium

Express in algebra: 'The product of two consecutive integers is 12 more than twice their sum.'

Example 3

medium

A rectangle's length is 3 more than twice its width

w

. Write the perimeter as an algebraic expression.

Example 4

medium

Translate Pythagoras's relation 'the square on the hypotenuse equals the sum of the squares on the other two sides' with legs

a,b

and hypotenuse

c

Example 5

hard

Read

\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}

as a sentence.

Example 6

challenge

Translate: 'the sum of the digits of

N = 10a + b

(two-digit) equals

S

' and prove

N - S

is divisible by 9.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write an equation: 'A number decreased by 7 equals 15.'

Example 2

medium

Express: 'The square of the sum of

a

and

b

' vs 'The sum of the squares of

a

and

b

'. Are they equal?

Example 3

easy

Translate 'five more than a number

x

' into algebra.

Example 4

easy

Translate 'the product of

a

and

b

' into algebra.

Example 5

easy

Write 'twice a number decreased by 7' as an expression.

Example 6

easy

Is the expression '

3 + \times 4

' grammatically valid algebra?

Example 7

easy

Translate the equation 'the sum of two consecutive integers is 15' using

n

Example 8

easy

Read the expression

\frac{a+b}{c}

in words.

Example 9

easy

Which symbol correctly expresses 'at least 10':

x>10

x\ge10

, or

x<10

Example 10

easy

Translate '

x

is between 2 and 8 inclusive' into a compound inequality.

Example 11

medium

The phrase 'the sum of

x

and

y

, squared' is ambiguous. Give both interpretations symbolically and which the grouping intends.

Example 12

medium

Translate the constraint 'a number is no more than twice another, and they sum to 12' into a system using

x,y

Example 13

medium

Why is

\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}

a grammar error analogous to a misread sentence?

Example 14

medium

Express 'the average of three numbers

a,b,c

equals 10' as an equation, then solve for

a+b+c

Example 15

medium

Read

\forall x\,(x^2\ge0)

in plain English and state whether it is true over the reals.

Example 16

medium

Translate 'John has 3 more coins than twice Mary's; together they have 21' into equations with

j,m

Example 17

medium

The notation

a < b < c

chains comparisons. Read it in words and explain why

a < b > c

is poor grammar.

Example 18

medium

Translate 'the square of the sum of

x

and 3 equals 49' into an equation and solve for

x

Example 19

medium

Read

|x| \le 3

in words and express it as a compound inequality without absolute value.

Example 20

challenge

Explain why

a/b/c

is ambiguous in the language of algebra, give both parsings, and state the standard convention.

Example 21

challenge

A student writes '

\sin x + 1 = \sin(x+1)

'. Diagnose the grammar error in terms of operator scope.

Example 22

challenge

Formalize 'there is a number whose square is itself' and find all such numbers, explaining the role of the existential quantifier.

Example 23

easy

Translate 'eight less than three times a number

n

' into an algebraic expression.

Example 24

easy

Translate 'the quotient of

a

and

b

' into algebra.

Example 25

easy

Translate 'a number is positive' into an inequality with variable

x

Example 26

easy

Translate 'the sum of three consecutive even integers' starting from

n

into an expression.

Example 27

easy

Translate 'half of a number decreased by 4' into algebra.

Example 28

medium

Translate 'the price

p

after a 15% discount' into an expression.

Example 29

medium

A taxi charges $3 plus $2 per mile. Write the cost

C

for

m

miles as an equation.

Example 30

medium

Translate: 'three times the difference of

x

and 5 equals four times

x

minus 6'. Then solve for

x

Example 31

medium

Read

(x+1)^2 - (x-1)^2

in words, then simplify.

Example 32

medium

Translate: 'twice the sum of a number and its reciprocal equals 5' with variable

x

(

x \ne 0

Example 33

medium

Read

f: \mathbb{R} \to \mathbb{R},\ f(x) = x^2

in plain English.

Example 34

medium

Translate 'no two distinct integers have the same square' as a logical statement.

Example 35

hard

A train leaves city A at speed

r

mph; 1 hour later, a second train leaves the same city at speed

r+10

mph. Translate 'the second train catches up' into an equation in time

t

(hours since first train left).

Example 36

hard

Translate 'the average of

x

and

y

is less than their geometric mean' into an inequality. For positive

x,y

, when is it true?

Example 37

hard

Translate: 'every positive integer can be written as a sum of four squares of non-negative integers'. Identify the structure (claim/quantifier shape).

Example 38

hard

A bag has

r

red and

b

blue marbles. Translate 'probability of drawing red equals one-third' into an equation.

Example 39

challenge

Translate Goldbach's conjecture into formal logic: 'every even integer greater than 2 is the sum of two primes'.

Example 40

challenge

A student writes '

\lim_{x \to 0} \dfrac{\sin x}{x} = \sin\dfrac{0}{0}

'. Diagnose the grammar/scope error.

Example 41

challenge

Express 'a function

f

is injective' in formal logic.

← Back to Algebra as Language Practice Mode

Related Concepts

Variables Expressions Mathematical Communication

Background Knowledge

These ideas may be useful before you work through the harder examples.

variablesexpressions