Introduction to Algebra
Algebra is a key part of GCSE Maths and forms the foundation for topics such as equations, graphs, functions, and formulas. This introduction to algebra explains how letters are used to represent numbers and how to solve simple equations step by step.
What is algebra?
Algebra is a branch of maths that uses letters (usually , , , , etc.) to represent numbers that may be unknown or can change.
Instead of working only with numbers, algebra lets us describe patterns, rules, and relationships.
For example:
Here, represents a number we don’t yet know.
Algebra is used to solve problems, describe situations, and build formulas across many areas of maths.
Example Question:
We will work through a simple algebra problem:
Find the value of :
Exam Tip
When solving algebra problems:
- Think of letters as numbers you haven’t found yet
- Use inverse operations to isolate the variable
- Keep the equation balanced by doing the same to both sides
Method – Solving a Basic Algebra Equation
We want to find the value of the unknown.
Step 1: Identify the operation
In , the number is being added to .
Step 2: Use the inverse operation
The inverse of is .
Subtract from both sides.
Step 3: Solve
Worked Example
Solve:
Step 1: Identify the operation
is being subtracted from .
Step 2: Use the inverse operation
Add to both sides.
Final answer:
Check your solution (important habit)
Substitute back into the original equation:
✔️
The solution is correct.
Reading algebra and substituting values
Letters represent numbers. A letter may be unknown, or it may take different values. These notation and substitution skills are used throughout Foundation and Higher GCSE.
Reading the notation
3a means 3 × a. ab means a × b. a² means a × a, while a³ means a × a × a. Thus a²b means a × a × b.
a/b means a ÷ b, provided b is not zero. The expression ½x means half of x. We often use fractional coefficients because they keep a value exact.
Brackets keep an expression together: 2(x + 3) means multiply the whole of x + 3 by 2. It is different from 2x + 3.
Useful words
An expression, such as 3x + 2, describes a value. An equation, such as 3x + 2 = 11, states that two values are equal.
A formula connects quantities: P = 4s connects a square’s perimeter P with its side length s.
The terms in 3x − 2 are 3x and −2. In 3x, the number 3 is the coefficient, and 3 and x are factors because they are multiplied.
An identity is true for every allowed value: 2(x + 3) ≡ 2x + 6. The symbol ≡ means “identically equal”. The equation x + 3 = 7 is true only when x = 4.
An inequality compares values: x < 5 means x is less than 5. The signs >, ≤ and ≥ mean greater than, less than or equal to, and greater than or equal to.
Worked example — Substitute a negative value
Find 2a² − b when a = −3 and b = 5. Substitution means replacing each letter with its given value.
2 × (−3)² − 5
Square first: (−3)² = 9.
2 × 9 − 5 = 18 − 5 = 13.
The brackets matter: we square the whole negative value. Do not calculate (2 × −3)², because only a is squared.
Worked example — Use a formula
A science question gives v = u + at. Here u is initial velocity, a is constant acceleration, t is time and v is final velocity.
Use u = 3 m/s, a = 2 m/s² and t = 4 s.
v = 3 + 2 × 4
v = 3 + 8 = 11 m/s.
Multiply before adding. On a calculator, enter the full expression with brackets around any negative input.
Worked example — Write an expression from words
A notebook costs £n. Three notebooks and a £2 pen cost 3n + 2 pounds.
There is no single numerical total until n is known. If n = 4, the total is 3 × 4 + 2 = £14.
Substitution with fractions, roots and scientific formulae
Worked example — Keep a fractional input exact
Find 3a² − 2a when a = −½. Replace every a, keeping the negative fraction in brackets:
3 × (−½)² − 2 × (−½) = ¾ + 1 = 7/4. Square before multiplying; subtracting a negative adds.
Worked example — A root and a value in standard form
For q = √(r + 5), with r = 4, first calculate inside the root: q = √9 = 3. Here √ denotes the non-negative square root.
A supplied formula is E = mc². Use m = 2 × 10⁻³ kg and c = 3 × 10⁸ m/s. Square the whole value of c: c² = 9 × 10¹⁶. Then E = (2 × 10⁻³)(9 × 10¹⁶) = 1.8 × 10¹⁴ J. This is substitution; the numerical index rules are taught in Number.
Worked example — A supplied motion formula
For constant acceleration, s = ut + ½at². Here s is displacement in metres, u is initial velocity in m/s, a is acceleration in m/s² and t is time in seconds. With u = 2, a = 3 and t = 4:
s = 2 × 4 + ½ × 3 × 4² = 8 + 24 = 32 m. Only t is squared in the second term.
Another supplied rule is v² = u² + 2as, where v is final velocity. With u = 3, a = 2 and s = 4, v² = 9 + 16 = 25. If v is specified as non-negative, v = 5 m/s; without a sign condition the algebra permits v = ±5. OCR names these motion formulae; this lesson teaches how to use a supplied rule, not a separate mechanics course.
Check your understanding — Find 2a² + a for a = ½, and q = √(r + 5) for r = 11.
2 × (½)² + ½ = ½ + ½ = 1.
q = √(11 + 5) = √16 = 4.
Check your understanding — Use s = ut + ½at² with u = 1, a = 2 and t = 3. Also find E from E = mc² when m = 4 × 10⁻³ and c = 2 × 10⁴.
s = 1 × 3 + ½ × 2 × 9 = 12 m when the motion inputs use the units above.
c² = 4 × 10⁸, so E = (4 × 10⁻³)(4 × 10⁸) = 1.6 × 10⁶ J when m and c use the units above.
Think Like a Mathematician
Algebra isn’t just about solving equations — it’s about understanding relationships.
The letter can represent:
- An unknown value
- A number that can change
- A general rule
For example:
Perimeter of a square =
Here, represents the length of one side.
Common mistakes to avoid
- Treating letters as objects instead of numbers
- Forgetting to apply operations to both sides
- Mixing up inverse operations
- Not checking the solution
Most algebra errors happen because the equation becomes unbalanced.
Try one yourself
Solve the equation:
First: Use the inverse operation
Then: Solve for
Subtract 7 from both sides to undo adding 7.
x + 7 − 7 = 12 − 7
x = 5
Check in the original equation: 5 + 7 = 12. Both sides are equal.
A correct solution makes both sides of the equation equal.
More practice — notation and substitution
1. Write a × a × b and 0.5 × x using shorter algebraic notation.
a × a is a², so a × a × b = a²b.
0.5 is one half, so 0.5 × x = ½x (or x/2).
2. Find 4p − 3 when p = 2.
Replace p with 2: 4 × 2 − 3.
Multiply first: 8 − 3 = 5.
3. Find 3a² + b when a = −2 and b = −5.
Write 3 × (−2)² + (−5).
(−2)² = 4, so 3 × 4 − 5 = 12 − 5 = 7.
4. A triangle has area A = ½bh. Its base b is 8 cm and its perpendicular height h is 5 cm. Find A.
A = ½ × 8 × 5.
Half of 8 is 4, so A = 4 × 5 = 20 cm².
5. Is 3(x + 2) ≡ 3x + 6 an identity? Is x + 2 = 6 an identity?
Expanding gives 3x + 6 for every x, so the first statement is an identity.
The second works only when x = 4. It is an equation, not an identity.
Frequently Asked Questions
What does the letter actually mean?
A letter represents a number.
Sometimes it’s unknown (like in equations), and sometimes it can vary (like in formulas).
Continue Learning
First lesson in Algebra. Start here if algebra is new to you.