Maths Explained
Without Magic

§ 01What Is Algebra? The Very Idea of Solving for x

Underneath every formula and every rule sits one idea that makes the rest of mathematics possible: we use a letter to stand in for a number we don't know yet. That's the whole trick. That's algebra.

Think back to primary school, a young child can solve the following without ever having heard the word "algebra":

You probably saw the answer, 3, right away. Your brain subtracted 1 from 4 without you consciously deciding to do it. Algebra just takes that same mental shortcut and writes it down formally, so it still works even when the numbers are too large, too messy, or too abstract to "just see."

The letter x is just the □ with a name. It represents one specific number that we are trying to find. Our job is to isolate x on one side of the equals sign.

The Golden Rule: The Balance Scale

Picture an equals sign as a balance scale: whatever sits on the left weighs exactly the same as what sits on the right. Change one side without touching the other, and the scale tips: the equation stops being true.

Which gives us the golden rule that everything else in algebra builds on:

It's not really a "rule" so much as basic logic: if two quantities are equal and you do the same thing to both of them, they're still equal afterward. That's just what equality means.

To solve 1 + x = 4, we want to get x on its own. x has 1 added to it, so we remove that 1 by subtracting it. But we must subtract 1 from both sides:

The 1 on the left cancels (1 − 1 = 0), leaving x = 3. Every algebraic manipulation you'll ever do, right up through calculus, is really just this same principle applied creatively.

§ 02Building Up — Interactive Exercises

Each exercise below increases slightly in difficulty. Type your answer, press Check. If you are stuck, use the Hint button first before looking at the full working.

Level 1 — Finding the Missing Number

We are not yet writing "solve for x" — we are just asking: what number goes in the blank? Your brain already knows how to do this. We are only learning to write it formally.

Level 2 — One-Step Equations with Proper Notation

Now we write x formally and show every line. The principle is identical to Level 1 — only the notation changes. Do not let the symbol intimidate you: x is still just a number waiting to be found.

Level 3 — Two-Step Equations

Now x has two things happening to it. The rule for untangling two operations: always undo addition or subtraction first, then undo multiplication or division. Think of it as peeling layers off in reverse order — last thing applied is first thing removed.

Level 4 — Variables on Both Sides

When x appears on both sides of the equation, we need one extra step at the beginning: collect all the x terms together on one side. It does not matter which side you choose — convention is to put x terms on the left — but either side works as long as you apply the golden rule.

Level 5 — Quadratic Equations

When x is squared (x²), the simple one-step approach no longer works. There is now the possibility of two answers, because both a positive and a negative number can be squared to give a positive result. For example: 5² = 25 and (−5)² = 25. Both values of x satisfy x² = 25. We always need to consider both.

§ 03Simultaneous Equations — Two Unknowns, Two Clues

A single equation with two unknowns has infinitely many solutions. Put two equations together, though, one for each unknown, and they usually pin down exactly one answer. Geometrically, that's just two lines crossing at a point.

Take the equation x + y = 7 on its own. It's satisfied by (1,6), (2,5), (3,4), (0,7), and infinitely many other pairs, there's no way to pin down unique values for x and y from one equation alone.

Now bring in a second equation, x − y = 1. Both conditions have to hold at the same time, and it turns out only one pair (x, y) can do that. Each equation is a straight line on a graph, and the solution is simply the point where those two lines meet.

Visualise It: Two Lines Crossing

Drag the sliders below and watch the intersection point move in real time. Parallel lines, the ones with the same slope, never cross at all, so that system has no solution.

Method: Elimination

The elimination method is the most reliable approach. The idea is to multiply one (or both) equations by a constant so that when you add or subtract them, one variable cancels out completely. You then have a single equation in one unknown, which you can solve with the techniques from the previous section.

Interactive Solver — Enter Your Own System

Enter any integer coefficients for ax + by = c. The solver shows the elimination working step by step.

§ 04Maths Is Not Magic — It Is Logic

You've probably heard that some people are just "maths people" and others aren't. It isn't true, and it never was, it's a myth that's talked plenty of capable students out of a subject they could have handled just fine.

Mathematics is a language, and you already know how to learn one of those, you're reading this sentence in one right now. Notation is a second language layered on top of the first, and like any language, it takes time, repetition, and, above all, someone explaining it properly.

When a student looks at x² + 5x + 6 = 0 and draws a total blank, it's rarely because the maths is genuinely beyond them. Usually a step got skipped somewhere along the way. A formula turned up without any explanation of where it came from. Notation appeared before anyone defined it. The lecture moved on before the idea had time to sink in.

That's the gap this site tries to close. Every formula gets derived before it's used, every symbol gets explained the moment it shows up, and you won't find the phrase "it can be shown that…" anywhere without the showing actually happening right below it.

The Problem with Most Maths Textbooks

Most textbooks are written by people who understood this material so long ago that they've forgotten what it felt like to be confused by it. Derivations get compressed. Words like "clearly" and "obviously" get dropped into spots that are neither, not to someone meeting the idea for the first time, anyway. The steps that feel trivial to an expert are usually exactly the steps a beginner needed spelled out.

Which is how you end up with a student who can follow a worked example line by line but can't reproduce it a week later, because they never actually grasped why one line led to the next. They'd memorised the shape of the answer, not the reasoning behind it. Change the wording of the question slightly, and they're stuck.

Speed Isn't the Point — Clarity Is

There's a common assumption that being good at maths means being fast at it. In practice, calculation speed has surprisingly little to do with real understanding. Some of the most important mathematicians in history were slow, deliberate thinkers who didn't rush anything.

Don't judge your progress by how quickly you get through a problem. Judge it by whether you can explain, in plain English, why each step has to happen. Manage that, and you understand it. If you can't, you've memorised a routine without the logic underneath it, and that kind of knowledge tends to fall apart the moment exam pressure hits.

Take your time here. Read explanations twice if you need to. Ask "why" at every step, and if this site doesn't answer it, that's a gap in the explanation, not in you. Let us know and we'll fix it.

§ 05All Chapters — Where to Go Next

This site covers the full varsity first and second year curriculum. Start from the beginning if you are new to calculus, or jump directly to the topic you need.