Algebra Basics: A Beginner’s Guide to Variables, Expressions and Equations

Algebra is math with a missing number. Instead of writing a blank, you use a letter such as x, then use a few simple rules to find what that letter stands for. This guide explains algebra basics in plain words: the key terms, the difference between an expression and an equation, how to simplify, and how to solve for x step by step.

In short: Algebra uses letters for unknown numbers. You simplify expressions by combining like terms, and you solve equations by doing the same thing to both sides until the letter is on its own.

What Is Algebra?

Algebra is the part of math that uses letters, called variables, to stand for numbers you do not know yet. If you know that a number plus 7 makes 12, you can write it as x + 7 = 12 and work out that x is 5.

The most useful idea in algebra is the balance. An equation is like a pair of scales: both sides weigh the same. As long as you do exactly the same thing to both sides, the scales stay level and the equation stays true.

In short: Algebra is arithmetic with an unknown number written as a letter. An equation is a balance, so whatever you do to one side, you do to the other.

Key Algebra Words You Need to Know

Here is every part of the equation 3x + 5 = 20, named:

What Is the Difference Between an Expression and an Equation?

An expression has no equals sign, so you cannot solve it. You can only simplify it or evaluate it for a given value. An equation has an equals sign, so you can solve it to find the value of the variable.

In short: No equals sign means it is an expression, so simplify it. An equals sign means it is an equation, so solve it.

How Do You Simplify an Algebraic Expression?

1. Combine Like Terms

Add or subtract terms that have the same variable part, and add the plain numbers separately.

Example: 4x + 3 + 2x − 1 → (4x + 2x) + (3 − 1) = 6x + 2.

2. Use the Distributive Property

A number outside brackets multiplies every term inside: a(b + c) = ab + ac.

Example: 3(x + 4) = 3x + 12. With a minus sign: 2(x − 5) − (x + 1) = 2x − 10 − x − 1 = x − 11.

3. Follow the Order of Operations

Work in this order: brackets, then powers, then multiplication and division from left to right, then addition and subtraction. You may know it as PEMDAS or BODMAS.

Example: 2 + 3 × 4 = 2 + 12 = 14, but (2 + 3) × 4 = 20. And 2 × 3² = 2 × 9 = 18, not 36.

In short: To simplify, open brackets with the distributive property, follow the order of operations, then combine like terms.

For more practice with substitution, powers and negative signs, read our guide to evaluating and simplifying algebraic expressions.

How Do You Solve a Simple Equation?

To solve for x, undo whatever is being done to x, one step at a time, using the opposite operation. Addition undoes subtraction, and division undoes multiplication. Always do the step to both sides.

One-Step Equations

Example: x + 7 = 12 → subtract 7 from both sides → x = 5.

Example: 4x = 28 → divide both sides by 4 → x = 7.

Two-Step Equations

1. Start with 3x + 5 = 20.
2. Subtract 5 from both sides: 3x = 15.
3. Divide both sides by 3: x = 5.

Undo the addition or subtraction first, then the multiplication or division. It is the order of operations in reverse.

Variables on Both Sides

Example: 5x − 4 = 2x + 11 → subtract 2x from both sides: 3x − 4 = 11 → add 4: 3x = 15 → divide by 3: x = 5.

Equations With Brackets

Example: 2(x − 3) = 10 → divide both sides by 2: x − 3 = 5 → add 3: x = 8. You could also expand first: 2x − 6 = 10, so 2x = 16 and x = 8.

In short: Undo each operation with its opposite, on both sides, in reverse order: deal with + and − first, then × and ÷, until x is alone.

Equations with fractions, decimals, negative answers or other letters are covered in our step-by-step guide to solving for x. To check any equation instantly, use the solve for x calculator.

How Do You Check Your Answer?

Put your answer back into the original equation. If both sides come out equal, your answer is right.

Example: For 3x + 5 = 20 with x = 5: 3 × 5 + 5 = 15 + 5 = 20. Both sides are 20, so x = 5 is correct.

You can do this check in the CalcSolver calculator: type 3×5+5 and press =. If you get 20, your answer works. CalcSolver is a numeric calculator, so it checks your answer rather than solving the letters for you, which is exactly what builds algebra skill.

In short: Substitute your answer into the original equation. If the two sides match, the answer is correct.

Basic Algebra Rules to Remember

• Keep the balance: whatever you do to one side of an equation, do to the other.
• Use opposite operations: + undoes −, and × undoes ÷.
• Multiply every term in brackets: a(b + c) = ab + ac.
• Only combine like terms: 2x + 3x = 5x, but 2x + 3 stays as it is.
• Watch the signs: subtracting a negative is adding, so 5 − (−2) = 7.
• Never divide by zero: it has no answer.

Common Beginner Mistakes

• Combining unlike terms. 2x + 3 is not 5x. The x term and the plain number must stay separate.
• Only multiplying the first term in brackets. 3(x + 4) is 3x + 12, not 3x + 4.
• Changing one side only. If you subtract 5 on the left, subtract 5 on the right too.
• Losing a minus sign. −(x + 1) is −x − 1, because the minus applies to both terms.

Practice Problems (With Answers)

Try each one, then open it to check. Substitute your answer to prove it.

Algebra Basics: Frequently Asked Questions

Sources and Further Reading

• OpenStax, Elementary Algebra 2e, Chapter 2: Solving Linear Equations and Inequalities, Rice University.
• Math is Fun, Introduction to Algebra.

What to Learn Next

1. Algebraic expressions: evaluate and simplify with confidence.
2. How to solve for x: every type of linear equation, step by step.
3. Functions in algebra: f(x), zeros, domain and range.

Every example in this guide was checked by substituting the answer on the CalcSolver calculator. Want quick mental shortcuts too? Read our 17 mental math tricks, or see the basic algebra formulas.