Algebra basics
Variables, expressions, equations — and the simple rules that make algebra a language rather than a puzzle.
1. Algebra is shorthand for arithmetic
Arithmetic deals with specific numbers: 3, 17, −0.5. Algebra is what you get when you swap one or more of those numbers for a letter that stands in for a number. The letter does not change the rules — it just lets you talk about a whole family of arithmetic facts at once. Where arithmetic might say "3 plus 5 is 8", algebra says "x plus 5 is x + 5", and that one expression instantly covers every value x could take.
Two ideas come up everywhere:
- An expression is something you can evaluate once you know the value of every variable.
3x + 7is an expression. If x = 2, the expression has value 13. - An equation is a claim that two expressions are equal.
3x + 7 = 16is an equation. The job of algebra is to find the value (or values) of the variable that make the claim true.
2. The four golden rules
Almost everything you ever do in algebra is one of four moves applied to both sides of an equation:
- Add the same thing to both sides.
- Subtract the same thing from both sides.
- Multiply both sides by the same non-zero thing.
- Divide both sides by the same non-zero thing.
Because the equation is a claim of equality, anything you do to the left must be done to the right too. Get this discipline early and the rest of algebra unfolds without much pain.
3. Solving a linear equation, line by line
Take 3x + 7 = 16. Subtract 7 from both sides: 3x = 9. Divide both sides by 3: x = 3. That is the entire procedure for a linear equation. The general form is ax + b = c, and the solution is always x = (c − b) / a. The equation solver will show every one of those steps.
4. Combining like terms
Before you can solve an equation you usually need to clean it up. Two terms are like if they have the same variable raised to the same power. 3x and 5x are like terms; you can add them to get 8x. 3x and 3x² are not like terms because the second has x squared; you cannot combine them. 3x and 5y are not like terms because the variables differ.
5. The distributive law
The single most useful identity in algebra is a(b + c) = ab + ac. It is what lets you expand brackets when needed and factor them out when convenient. You also need its mirror: a(b − c) = ab − ac, and the more general FOIL pattern for two-binomial products: (a + b)(c + d) = ac + ad + bc + bd.
6. Powers and indices
The index laws are short, fixed, and worth memorising:
x^a · x^b = x^(a + b)x^a / x^b = x^(a − b)(x^a)^b = x^(a · b)x⁰ = 1for any non-zero xx^(−a) = 1 / x^a
7. Common pitfalls
Three mistakes account for most algebra slip-ups in early secondary school:
- Forgetting to do something to both sides. If you subtract 5 from the left, subtract 5 from the right.
- Distributing a minus sign.
−(x + 3)is−x − 3, not−x + 3. - Confusing addition and multiplication of like terms.
x + x = 2x, butx · x = x².
8. What to read next
Once linear equations feel comfortable, move on to quadratic equations — the next big jump in algebra, and the one that introduces the discriminant, the parabola, and complex numbers.