Algebra

(solving for the unknown: arithmetic extended to work with variables)

What is algebra?

Algebra takes the rules of arithmetic and applies them to numbers we don't know yet. Instead of writing a specific number, we use a letter — usually x — to stand in for "the number we're trying to find." Algebra is really just arithmetic with a placeholder.

Expressions and equations

An expression is just a combination of numbers, letters, and operations, like 3x + 5. It doesn't claim anything is equal to anything else — it's just a recipe. An equation says two expressions are equal, like 3x + 5 = 20. Solving an equation means finding the value of x that makes it true.

Solving an equation: the balance scale idea

Think of an equation as a balance scale that is perfectly level. Whatever you do to one side, you must also do to the other side — otherwise the scale tips, and the equation stops being true. This one idea is the entire foundation of solving equations.

Let's solve 3x + 5 = 20, step by step, checking that the scale stays balanced at every step.

Step 1: subtract 5 from both sides. 3x + 5 − 5 = 20 − 5, which simplifies to 3x = 15. We're allowed to do this because we removed the exact same amount from both sides — if the two sides were equal before, they're still equal now.

Step 2: divide both sides by 3. 3x ÷ 3 = 15 ÷ 3, which simplifies to x = 5. Again, dividing both sides by the same number keeps them equal.

So x = 5. We can check this is correct by putting 5 back into the original equation: 3×5 + 5 = 15 + 5 = 20. It matches, so the answer is confirmed correct — not just "probably right."

Why these steps are always allowed

More generally: for any equation, you may add the same number to both sides, subtract the same number from both sides, multiply both sides by the same number, or divide both sides by the same number (as long as it isn't 0) — and the equation will still be true afterward. This is guaranteed, not just observed, because equality means "these two amounts are exactly the same," and performing the identical action on two amounts that are the same always produces two results that are still the same as each other.

A first look at quadratic equations

Some equations involve x multiplied by itself (written x², "x squared"). These are called quadratic equations, and they can have two solutions instead of one. For example, x² − 5x + 6 = 0 can be solved by rewriting the left side as (x − 2)(x − 3) — you can check this is the same thing by multiplying it back out. Since two numbers multiply to give 0 only when at least one of them is 0, either x − 2 = 0 (so x = 2) or x − 3 = 0 (so x = 3). Both are genuine solutions: putting either one back into the original equation makes it true.

Why algebra matters

Algebra lets us describe a relationship in general — "twice a number, plus 3" — rather than only being able to talk about specific numbers one at a time. That single shift is what makes it possible to solve problems where the answer isn't known in advance, from splitting a bill fairly to working out how fast something is moving.