LESSON 12
VARIABLES & EXPRESSIONS
The Language of Algebra
What is a Variable?
A variable is a letter or symbol that represents an unknown or changing value. Variables allow us to write general rules and formulas that work for any number, not just one specific case. The most commonly used variables are x, y, n, and a.
An algebraic expression is a combination of variables, numbers, and operations. Unlike an equation, an expression does not have an equals sign — it simply represents a value. Learning to read, write, and simplify expressions is the foundation of all algebra.
Writing Algebraic Expressions
Mathematical operations can be written in words and translated into algebraic expressions. 'The sum of a number and 5' becomes x + 5. 'Three times a number' becomes 3x (the multiplication sign is dropped when a number is next to a variable). 'A number divided by 4' becomes x/4.
Evaluating Expressions
To evaluate an expression means to find its value when the variable is replaced by a specific number. Substitute the given value in place of the variable, then simplify using the order of operations. Always use parentheses when substituting to avoid sign errors.
Like Terms
Like terms are terms that have the same variable raised to the same power. Only like terms can be combined by adding or subtracting their coefficients. The coefficient is the number in front of the variable. Constants (plain numbers with no variable) are also like terms with each other.
The Distributive Property
The distributive property states that multiplying a number by a group in parentheses is the same as multiplying it by each term inside. This is one of the most used tools in algebra for expanding and simplifying expressions.
Key Facts
The word 'algebra' comes from the Arabic 'al-jabr', from a 9th-century book by mathematician Al-Khwarizmi, who is considered the father of algebra.
A coefficient is the number multiplied by a variable. In 7x, the coefficient is 7. If no number is written, the coefficient is 1 — so x means 1x.
Variables were not always letters. Ancient mathematicians used words or geometric lengths to represent unknowns before symbolic notation was developed.
The distributive property works in reverse too — factoring. 6x + 9 = 3(2x + 3). Recognizing this pattern is key to solving equations efficiently.