Algebra foundations · 01 · First moves · 7 min
Variables & expressions
Algebra begins with one brave act: giving a name to a number you don't know yet. Call it x, and suddenly you can write down everything you do know about it — and push.
Build the intuition
x is a name, not a mystery
“A number, doubled, then increased by three” becomes 2x + 3. Nothing is hidden or unsolved — x is simply the name of the number we're talking about, the way “the suspect” names a person before you know who they are.
Expressions are recipes
An expression is a recipe waiting for an ingredient. Feed x = 5 into 2x + 3 and out comes 13. Feed in 100, out comes 203. One short string of symbols handles every number that will ever exist — that compression is algebra's whole superpower.
Like terms: collecting the herd
3x + 2x is 5x for the same reason three apples plus two apples is five apples. Simplifying an expression is just herding identical things together — the expression shrinks, the meaning doesn't change.
See it move
An expression is a machine: numbers in, numbers out, same rule every time. Feed it inputs and watch the recipe run.
A worked example
From words to symbols
A taxi charges a $4 base fee plus $2 per kilometer. Write the cost of a ride.
Name the unknown: let k be the kilometers driven.
Translate the sentence:
Now every possible ride is answered at once: 5 km costs 2(5) + 4 = 14. The formula is the fare table — all of it.
Out in the world
Spreadsheets are algebra farms
Every spreadsheet cell formula — =B2*1.21, =SUM(A1:A12) — is an expression over variables. Accountants, scientists, and planners do algebra all day; the grid just hides the x's.
Common confusion, cleared
“I need to find what x is.”
Not yet! Expressions describe; equations (next lesson) demand. 2x + 3 isn't a question — it's a perfectly complete recipe.
“2x means twenty-something.”
Side-by-side means multiply: 2x is 2 × x. Algebra drops the × sign because it looks too much like the letter x.
Recap
- A variable names an unknown or any number at once.
- An expression is a reusable recipe: 2x + 3 answers infinitely many cases.
- Combine like terms to simplify without changing meaning.