Algebra › Introduction

Introduction to Algebra

Algebra is arithmetic with names. When you don't know a number yet, you call it a letter — and the rest of the rules still work.

Substitute and evaluate

Pick an expression, slide the values, watch it become a number.

Step 1 — write the expression

2x + 3

Step 2 — replace each letter with a number

2(4) + 3

Step 3 — work it out

= 11

Substitution is the simplest move in algebra: erase the letter, write a number in its place, then evaluate normally.

The one big idea

Imagine a box with some apples in it, plus 3 more on top. The total comes to 8 apples. How many were in the box?

You can guess: it must be 5. But you can also write down the question using a letter for the unknown — say x for the number inside the box:

x + 3 = 8

That sentence is an equation. Algebra is the toolkit for finding the value of x without guessing.

Why a letter?

A letter is a placeholder. It says: "Whatever this number is, treat it the same way you'd treat any number." That little move unlocks a huge amount of math — the same equation now works for any starting amount, not just one.

x is just a number you haven't named yet

x has no magic. It obeys the same rules as 7 or 100. You can add to it, multiply it, square it — and once you know what it stands for, you can replace it back.

The vocabulary

Variable — a letter that stands for an unknown number, like x or n.
Constant — a number that doesn't change, like 3 or 100.
Coefficient — the number multiplied by a variable. In 4x, the 4 is the coefficient.
Term — a single piece. 3x, −5, y² are all terms.
Expression — terms joined by + or −. Example: 3x + 2.
Equation — two expressions joined by an equals sign. Example: 3x + 2 = 11.

Quick check

In the term 4x, which part is the coefficient?

Quick check

Which of these is an equation (not just an expression)?

Hidden multiplication

When a number sits next to a letter, that's a multiplication — even though the × is invisible. So 4x means 4 × x, and 3(x + 1) means 3 × (x + 1).

Why hide the ×? Because in algebra it would clash with the letter x itself. Skipping the symbol keeps everything readable.

Try this

What does 3(x + 2) mean, fully written out?

Remember the invisible multiplication.

What you can do next

Try plugging numbers in for letters — that's substitution.
Combine and simplify — that's expressions.
Solve for the unknown — that's equations.