What Is Algebra?
Algebra is a branch of Mathematics where we use letters (called variables) to represent numbers. Instead of writing a fixed number, we write a letter that can stand for different values.
You will use algebra in almost every topic in Secondary E-Maths and A-Maths — from equations and graphs to trigonometry and calculus. Building a strong foundation now will make harder topics much easier later.
Definition — Variable
A variable is a letter (such as x, y, or n) used to represent an unknown or changing value. For example, in the expression 3x + 5, the letter x is the variable.
Terms and Expressions
An algebraic expression is a combination of variables, numbers, and operations (such as + and −). It does not have an equals sign.
Each part of an expression separated by + or − is called a term:
- 3x — one term (3 is the coefficient, x is the variable)
- 5y² − 2y + 7 — three terms
- 4ab — one term with two variables a and b
Definition — Coefficient
The coefficient is the number in front of a variable. In 7x, the coefficient is 7. In −3y, the coefficient is −3. If there is no number written, the coefficient is 1 (so x means 1x).
Like Terms and Simplification
Like terms are terms that have exactly the same variable(s) raised to the same power. You can only add or subtract like terms.
Examples of combining like terms:
- 3x + 5x = 8x
- 7y − 2y = 5y
- 4a + 3b − a + 2b = 3a + 5b (group a terms and b terms separately)
- 2x² + 5x − x² + 3x = x² + 8x (x² and x are not like terms)
Substitution
Substitution means replacing a variable with a given number. This lets you evaluate (find the value of) an expression.
How to substitute:
- Identify the variable in the expression.
- Replace every instance of that variable with the given number.
- Follow BODMAS/BIDMAS to calculate the result.
Example:
If p = 4 and q = −2, find the value of 3p − q².
3(4) − (−2)² = 12 − 4 = 8
Expanding Brackets
Expanding means multiplying each term inside the bracket by the term outside it. This removes the brackets.
Single bracket:
- 3(x + 4) = 3x + 12
- −2(y − 5) = −2y + 10 (be careful with the negative sign)
- x(2x − 3) = 2x² − 3x
Double brackets (A-Maths level):
Use FOIL — First, Outer, Inner, Last:
- (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6
- (2x − 1)(x + 4) = 2x² + 8x − x − 4 = 2x² + 7x − 4
Factorisation Basics
Factorisation is the reverse of expanding — you take out a common factor from each term and write it outside a bracket.
How to factorise:
- Find the Highest Common Factor (HCF) of all the terms.
- Write the HCF outside the bracket.
- Divide each term by the HCF to find what goes inside the bracket.
Examples:
- 6x + 9 = 3(2x + 3) — HCF is 3
- 4x² − 8x = 4x(x − 2) — HCF is 4x
- 15a²b + 10ab² = 5ab(3a + 2b) — HCF is 5ab
Common Mistakes to Avoid
- Adding unlike terms: 3x + 4y ≠ 7xy. These cannot be simplified further.
- Wrong sign when expanding: −3(x − 2) = −3x + 6, not −3x − 6.
- Forgetting brackets when substituting negatives: if x = −2, then x² = (−2)² = 4.
- Not taking out the full HCF: 4x² + 6x — students often only take out 2 instead of 2x.
Practice Questions
- Simplify: 5x + 3y − 2x + y
- If a = 3 and b = −2, find the value of 2a² − 3b.
- Expand and simplify: 4(2x − 3) − 2(x + 5)
- Expand: (x + 5)(x − 2)
- Factorise completely: 12x²y − 8xy²
Secondary E-Math and A-Math Tuition Support
Once you are confident with algebra basics, you will use these skills in:
- Linear equations — solving for x
- Quadratic equations — factorising to solve x² + bx + c = 0
- Indices and surds — applying algebra to powers and roots
- Coordinate geometry — using algebra to describe straight lines and curves
StrongWill Tuition helps students strengthen algebra, equations and problem-solving skills through secondary E-Math and A-Math tuition in Singapore. View our secondary math tuition page or check our tuition fees.
If you need help with any of these topics, StrongWill Tuition provides structured Secondary Maths tuition that builds from algebra fundamentals right through to O-Level and A-Level exam preparation.
