Basics of Algebra — Variables, Expressions, Equations

Basics of Algebra — Variables, Expressions, Equations

Algebra

TL;DR

The basics of algebra are five objects — variables, constants, expressions, equations, and inequalities — combined by the four arithmetic operations under one balance rule. This article walks through each foundational concept with three worked examples, the most common beginner errors, and the 9th-century origin of the word algebra itself.

BT
Last updated on May 27, 2026

Around 820 CE, a scholar in Baghdad's House of Wisdom wrote a book whose Arabic title contained the word al-jabr — meaning "restoration" or "the reunion of broken parts." Twelve centuries later, that word names the entire branch of mathematics this article is about. The scholar was Muḥammad ibn Mūsā al-Khwārizmī, and his book Kitāb al-Jabr wa-l-Muqābala described exactly the procedures students still call algebra today — moving terms across an equals sign, balancing both sides, solving for the unknown.

The basics of algebra are the small set of objects (variables, constants) and operations (addition, subtraction, multiplication, division, exponents, equals) you use to write down relationships involving unknown quantities and solve for them. Everything from a Grade 6 word problem to differential calculus is built on these five ideas.

What Is Algebra?

Algebra is the branch of mathematics that uses letters and symbols to represent numbers, quantities, and the relationships between them. Where arithmetic answers "what is 3+5?", algebra answers "if x+5=12, what is x?"

The five concepts that make up the basics of algebra:

  1. Variables — letters that stand in for unknown numbers (x, y, n).

  2. Constants — fixed numerical values (7, -12, π).

  3. Expressions — combinations of variables and constants joined by operations (3x+5, ab-2).

  4. Equations — statements that two expressions are equal (3x+5=14).

  5. Inequalities — statements that one expression is greater or less than another (2x+1≤9).

To "do algebra" is to apply legal moves — adding the same thing to both sides, factoring, distributing — that preserve the truth of the statement until the unknown stands alone.

The Building Blocks in Detail

Variables And Constants

A variable holds a value you don't yet know — or a value that can change. In x+3=10, x is the unknown; you do work to discover that x=7. In the formula A=πr², the radius r is a variable because it varies from circle to circle.

A constant does not change. In 3x+5, the 3 and the 5 are constants (specifically, 3 is the coefficient of x, and 5 is a free constant term).

Expressions vs Equations

An expression is a phrase: 3x+5. No equals sign, so you can't "solve" it — only evaluate it or simplify it.

An equation is a sentence: 3x+5=14. The equals sign asserts that the two sides are the same number. Solving the equation means finding the value (or values) of x that make this assertion true.

Operations And The Order They Follow

Algebra inherits the four arithmetic operations — addition, subtraction, multiplication, division — and adds exponents and roots. The order of operations (PEMDAS in the US, BODMAS in the UK) tells you what to evaluate first: parentheses, exponents, multiplication and division, addition and subtraction.

The Five Algebra Rules Every Student Needs

Rule What it says Example
Commutative Order doesn't matter for + and × 3 + 7 = 7 + 3; 2x⋅5 = 5⋅2x
Associative Grouping doesn't matter for + and × (2 + 3) + 4 = 2 + (3 + 4)
Distributive Multiplication distributes over + and − 3(x + 4) = 3x + 12
Identity Adding 0 or multiplying by 1 changes nothing x + 0 = x; x⋅1 = x
Inverse Every a has −a and (if a≠0) 1/a 5 + (-5) = 0; 5⋅(1/5) = 1

These five rules — together with "whatever you do to one side, do to the other" — are the entire grammar of school algebra.

Three Worked Examples — Quick, Standard, Stretch

Quick. Solve x+7=12.

Subtract 7 from both sides: x=12−7=5.
Final answer: x=5.

Standard (The Mistake Worth Making Once). Solve 3x−4=2x+11.

The wrong path. A student thinks "I'll move the 4 over first" and writes 3x=2x+11+4, then divides everything by x: 3=2+15x, which gives 15x=1 and x=15. Plug back in: 3(15)−4=41 and 2(15)+11=41 on the right. The number happens to be right, but the reasoning was broken.

The rescue. Collect the variable terms on one side and the constants on the other. Subtract 2x from both sides: x−4=11. Add 4 to both sides: x=15. Final answer: x=15.

Stretch. Solve the literal equation ax+b=c for x, where a,b,c are constants and a≠0.

Subtract b from both sides: ax=c−b. Divide both sides by a: x=c−b/a. Final answer: x=c−b/a, valid for any a≠0. This is the general solution of a linear equation in one variable.

Why Algebra Matters — The Real-World Pay-Off

Algebra is the language every quantitative field speaks. The substance of algebra — using a symbol to stand for an unknown so you can reason about it — is what makes science, engineering, and economics possible.

Slip-Ups That Cost Marks on Basics of Algebra

Mistake 1: Treating the equals sign as a "do this now" command.

Where it slips in: Younger students see 3+4= as a prompt to compute. The correct way: The equals sign is a balance, not an arrow.

Mistake 2: Distribution errors with negative signs.

The correct way: The negative sign distributes across every term inside the parentheses.

Mistake 3: Combining unlike terms.

The correct way: Like terms share the same variable to the same power.

The People Behind Algebra

Conclusion