Introduction to Algebra

Algebra is one of the most important branches of mathematics. Its essence is using letters (variables) instead of or together with numbers. This enables the description of general rules and relationships, not just specific calculations.

What is a Variable?

A variable is a symbol that represents an unknown or arbitrary value. Most commonly denoted by letters x, y, or z.

In this equation, x is the unknown. If x = 4, the equation holds: 4 + 3 = 7.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations. For example, in the expression 3x + 5, 3 is the coefficient of the variable, x is the variable, and 5 is a constant.

• Constant: fixed number, e.g., 5.
• Variable: symbol, e.g., x.
• Coefficient: the number before the variable, e.g., 3 in 3x.

Rules of Basic Operations

Algebraic operations follow the same rules as operations with numbers, supplemented by some important properties.

• Commutativity: a + b = b + a
• Associativity: (a + b) + c = a + (b + c)
• Distributivity: a(b + c) = ab + ac

Solving Equations

An equation holds when the left and right sides have the same value. We can find the unknown value using various transformations.

Solution: subtract 1 from both sides (2x = 6), then divide by 2 (x = 3).

Polynomials

A polynomial is an algebraic expression consisting of multiple terms where variables appear only in positive integer powers. Example: x² − 5x + 6.

This is an example of factoring: the polynomial is broken down into the product of two simpler factors.

Algebra in Everyday Life

• If a product's price is x forints and you buy two, the total cost is 2x.
• If a car travels y km/h, in 3 hours it covers 3y km.
• If you have 10 candies and give away z, you have 10 − z left.

Practice Exercise