PURE ALGEBRA · TEXTBOOK LESSON
Algebra uses symbols to represent numbers and relationships. The central principle is equivalence: every transformation must preserve the value or solution set represented by the original expression or equation.
Learning objectives
By the end of this lesson, you should be able to:
- Use algebraic notation correctly.
- Simplify expressions using valid algebraic laws.
- Solve equations by performing equivalent operations.
- Substitute values into expressions and formulae.
- Check solutions by substitution.
1. Key vocabulary and definition
Definition
An expression contains numbers, variables and operations but no equality sign. An equation states that two expressions are equal. Solving an equation means finding values that make the equality true.
variablecoefficienttermexpressionequationidentitysolution
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- Only like terms can be combined by addition or subtraction.
- Whatever operation is performed to one side of an equation must preserve equality.
- The distributive law is a(b+c)=ab+ac.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Simplify both sides where possible.
2
Use inverse operations to isolate the unknown.
3
Keep the equation balanced at every step.
4
For products, factors or quadratics, choose an appropriate algebraic technique.
5
Substitute the solution back into the original equation to check.
4. Worked example
Worked example
Solve 3x + 5 = 23.
1
Subtract 5 from both sides: 3x = 18.
2
Divide both sides by 3: x = 6.
3
Check: 3(6) + 5 = 18 + 5 = 23.
4
Therefore x = 6 is the solution.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- Simplify 4a + 3a - 2.
- Solve 5x - 7 = 18.
- Expand 3(2x + 5).
- Factorise 6x + 15.
- If y = 2x² - 3, find y when x = 4.
Show answers
- 7a - 2
- x = 5
- 6x + 15
- 3(2x + 5)
- 29
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.