This is where most people quit — the spot where a world of numbers suddenly turns into symbols. But x was not invented to make things harder; it was invented to make them easier. Saying so convinces nobody, so here are two things you can work with your own hands.
First handle — the equals sign
School trained us to put an answer after every =. So 3+x=11 jams: the answer slot is already occupied. But the real meaning of = is a declaration that both sides are the same. It is a balance scale. Tip it yourself.
What you just did is the whole of "solving an equation" — finding the x that levels the scale. Moving terms across, cancelling: those are just shortcuts for doing this one thing faster. Once = looks like a scale, the shortcuts can wait.
Second handle — generality
Pick a number. Double it, add 4, halve it, subtract the number you started with. Drag the slider as much as you like — the last line never moves. The numbers will not tell you why.
Feed it a hundred numbers and they only ever say "2 again". The letters answer in one line — whatever x is, halving gives x+2, and subtracting x leaves 2. One line standing in for infinitely many checks. That is why letters exist. x is not one unknown number — it is a hand holding every number at once.
This agreement took three thousand years
"A quantity and its quarter, added together, make fifteen" — here is that single problem, written the way each age wrote it.
"Aha and a quarter of aha make 15"
The Ahmes papyrus called the unknown "aha" — a heap. Every problem was prose, so every solution was prose too.
"The unknown and its quarter = 15"
Diophantus of Alexandria first gave the unknown a shorthand mark, and the word algebra comes from al-Khwarizmi's book title al-jabr — a book which, for all that, contained no formulas at all. Only sentences.
x + x/4 = 15
Writing unknowns with the last letters of the alphabet — x, y, z — settled in the 17th century. Once a problem fit on one line, solving became handwork. A three-thousand-year agreement is now something you are expected to absorb in a term.
So if you were stuck here for a long time, nothing is wrong with you — humanity as a whole was stuck here for three thousand years.
Names for the things you just worked with, and where the notation came from.
Browse the full glossary →There are no exercises here. If you saw the scale come level once, this page has done its job. The rest belongs not to a workbook but to the next node on the map.
If one letter x can hold every number at once — what would two letters, x and y, hold? That answer is the map's next cliff: functions.