Variations & Expressions

Algebra begins with variables, the symbols (usually letters) that stand in for unknown or changing quantities. Together with constants and operations, they form expressions – mathematical phrases like 2x + 5 that encapsulate relationships without yet declaring equality. Learning to read and write expressions fluently lets you translate word problems into compact algebraic form.

In practice, you might see an expression such as 3a – 2b + 7. Here, a and b could represent apples and bananas in a fruit basket, and the expression calculates a combined value. Understanding that 3a means “three times the amount of a” is crucial to tackling puzzles that involve proportional reasoning or symbolic manipulation.

Example: Evaluate 2x + 5 when x = 3.

Substitute: 2·3 + 5

Multiply: 6 + 5

Add: 11
So, when x = 3, the expression equals 11.

Hints and tips

• Choose meaningful letters: Use c for cost, t for time, n for number of items – to make your work more readable.
• Keep constants separate: Write numbers (e.g. 5, 12) apart from variables so you don’t accidentally combine them.
• Track units mentally: If x represents “kilograms,” remind yourself “3x” means “3 kg times x.”
• Always substitute carefully: When plugging in a value, rewrite the entire expression first (e.g. replace all x with 4 before calculating).
• Check domain restrictions: If a variable must be positive or an integer, note that before solving complex problems.