Introduction to Algebra

What you'll learn

Understand the concept of variables and constants
Form and simplify algebraic expressions
Solve simple one-step equations
Recognise and extend number and shape patterns

Key concepts

Variables and Constants

Algebra uses letters to represent unknown or changing quantities.

Term	Definition	Examples
Variable	A letter representing an unknown or changing value	x, y, n, a, b
Constant	A fixed number that never changes	3, −7, 100, ½
Coefficient	The number multiplied with a variable	In 5x, the coefficient is 5
Term	A single number, variable, or product of both	4x, −3, 7y², 2

Examples of variables in real life:

Number of students in a class = n (can change each year)
Price of a pen = p (varies by brand)
Age of a person = a (changes each year)

Worked Example: Sana has 3 more pencils than Ravi. If Ravi has x pencils, Sana has x + 3 pencils.

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and operations (no equals sign).

Parts of an expression:

For the expression 3x² + 5x − 7:

Component	Value
Terms	3x², 5x, −7
Coefficients	3 (of x²), 5 (of x)
Constant term	−7
Number of terms	3 (trinomial)

Types of expressions by number of terms:

Type	Terms	Example
Monomial	1	4x, −3y², 7
Binomial	2	2x + 5, a − b
Trinomial	3	x² + 3x − 2
Polynomial	Many	x³ − 2x² + x − 4

Like and Unlike terms:

Like terms	Unlike terms
Same variable, same power	Different variables or powers
3x and 7x	3x and 7y
−4y² and 9y²	4x² and 4x
5ab and −2ab	5a and 5ab

Simplifying expressions — add/subtract like terms only:

Worked Example: Simplify 4x + 3y − 2x + 5y Group like terms: (4x − 2x) + (3y + 5y) = 2x + 8y

Worked Example: Simplify 7a − 3b + 2a + b − 4 = (7a + 2a) + (−3b + b) − 4 = 9a − 2b − 4

Adding expressions:

(2x + 5) + (3x − 1) = 5x + 4

Substitution — replacing a variable with a number:

Worked Example: Find the value of 3x − 4 when x = 5. 3(5) − 4 = 15 − 4 = 11

Worked Example: Evaluate 2a² + b when a = 3, b = −1. 2(9) + (−1) = 18 − 1 = 17

Simple Equations

An equation has an equals sign (=). It says two expressions are equal.

Expression vs Equation:

Expression	Equation
2x + 3	2x + 3 = 9
No equals sign	Has equals sign
Cannot be solved	Can be solved

Key idea — Balancing: Whatever you do to one side, do the same to the other side.

Solving one-step equations:

Operation needed	Example	Solution
Subtract from both sides	x + 4 = 10	x = 10 − 4 = 6
Add to both sides	x − 3 = 7	x = 7 + 3 = 10
Divide both sides	5x = 20	x = 20 ÷ 5 = 4
Multiply both sides	x/3 = 6	x = 6 × 3 = 18

Worked Example: Solve 2x + 3 = 11 Step 1: Subtract 3 from both sides → 2x = 8 Step 2: Divide both sides by 2 → x = 4 Check: 2(4) + 3 = 11 ✓

Worked Example (word problem): Priya thinks of a number. She triples it and adds 5, getting 20. Find the number. Let the number = n 3n + 5 = 20 3n = 15 n = 5

Verification: Always substitute back to check your answer.

Patterns

Patterns are regular arrangements of numbers or shapes following a rule.

Number patterns:

Pattern	Rule	Next 3 terms
2, 4, 6, 8, …	Add 2	10, 12, 14
3, 6, 12, 24, …	Multiply by 2	48, 96, 192
1, 4, 9, 16, …	Perfect squares (n²)	25, 36, 49
1, 3, 6, 10, …	Triangular numbers (+2, +3, +4…)	15, 21, 28
1, 1, 2, 3, 5, 8, …	Add two previous terms (Fibonacci)	13, 21, 34

Finding the rule (nth term):

Position (n)	1	2	3	4	5
Value	3	6	9	12	15

Rule: Multiply position by 3 → nth term = 3n

Worked Example: Find the 10th term of the sequence 5, 8, 11, 14, … Pattern: starts at 5, add 3 each time. nth term = 5 + (n − 1) × 3 = 3n + 2 10th term = 3(10) + 2 = 32

Shape patterns and algebra:

Matches needed to build n squares in a row:

Squares (n)	Matches
1	4
2	7
3	10
n	3n + 1

Formula: 3n + 1 matches are needed for n squares.

Quick check

Identify the variables, coefficients, and constant in: 4x² − 3x + 7
Simplify: 6a − 2b + 3a + 5b − 1
If p = 4 and q = −2, evaluate 3p − q²
Solve: 4x − 7 = 13
The sequence 2, 5, 8, 11, … What is the 15th term?

Open the Practice tab for graded questions on Introduction to Algebra.

Algebra Intro