Introduction to Variables

A bill that isn’t one number

Your phone plan costs $$20$ a month, plus $$3$ per gigabyte of data. What’s the bill? There’s no single answer — it depends on the data. But the pattern is fixed, and you can write it down once: $20 + 3g$, where $g$ stands for however many gigabytes you use. That little letter is the entire leap into algebra. A variable is a placeholder for a number you don’t know yet — and an expression built around one isn’t a question waiting for an answer, it’s a machine: feed in any month’s usage, out comes that month’s bill.

The machine runs on grammar you already own: $20 + 3g$ obeys the exact same priority ladder as F1’s expressions — multiply $3$ by $g$ first, then add — the only novelty is a blank where one number used to be.

The vocabulary, from the phone bill

Each piece of an expression has a name, and the bill makes them concrete. A term is one chunk glued by multiplication: $20$ and $3g$ are the two terms here. A coefficient is the number riding a variable — the $3$ in $3g$, and it means something: three dollars per gigabyte. A constant is a term with no variable — the $20$, the part of the bill that never changes. Almost everything in this unit is tidying expressions built from these pieces.

Evaluating — feed the machine

Suppose you used $4$ gigabytes. Evaluate the expression by substituting the value and computing with the usual order of operations: $20 + 3(4) = 20 + 12 = 32$ dollars. That’s all “plugging in” is — the moment a variable’s value is known, the whole expression collapses to a number.

The widget opens on $3x + 2$ with $x = 4$. Predict the result, then check. Now try the same expression with $x = -1$ — careful, the coefficient multiplies the whole $-1$ — and then type $20 + 3x$ with your own “data usage.”

Like terms — you can only count matching kinds

Simplify $3x + 2 + 5x - 6$. The $x$-terms count together ($3$ $x$‘s plus $5$ $x$‘s is $8x$), the constants count together ($2 - 6 = -4$), and the answer is $8x - 4$. What you may not do is merge $8x$ with $-4$ into one number — and the itch to do it anyway is worth understanding. Arithmetic spent years training you that a finished answer is a single number, so $3x + 5$ feels unfinished, and "$8x$" scratches the itch. But $x$-terms and constants are different units — $3x$ means “three $x$‘s,” the way $3$ inches can’t merge with $5$ miles (the same unit-thinking from F3). The lie shows up the moment a value arrives: at $x = 2$, $3x + 5$ is $11$, while $8x$ would be $16$. An expression like $8x - 4$ is a finished answer — a number-in-waiting.

The distributive property — multiply across

You already distribute in your head. Asked for $7 \times 103$, you’d never stack the long multiplication — you’d split it: $7 \times 100$ plus $7 \times 3$, so $721$. Letters just make the split official: a number outside parentheses multiplies every term inside, $a(bx + c) = abx + ac$. The picture is a rectangle’s area: height $a$, width split into $bx$ and $c$ — the total area is the two panels added.

The box opens on $5(2x - 3)$. Predict both terms of the answer before you look, then make the outside number negative and watch what happens to each panel of the rectangle.

The negative-sign trap

That last experiment is the single most costly slip in early algebra. In $-2(x + 4)$, the outside factor is the whole $-2$ — sign included — and it must reach every term: $-2x - 8$, not $-2x + 4$. Stopping after the first term feels natural because the eye reads ”$-2$ times $x$… done” and the $+4$ looks like it was already handled by its own plus sign. A five-second lie detector: substitute $x = 1$. The original gives $-2(5) = -10$; the wrong version gives $-2 + 4 = 2$; the right one gives $-2 - 8 = -10$ ✓. That trick — test any simplification with a small number — catches nearly every algebra slip you’ll ever make.

Put it together: distribute, then combine

Real problems mix the two skills: distribute every set of parentheses first (watching signs), then count the like terms. Give the widget $5(2x - 3) - 2(x + 4)$ and predict just the $x$-coefficient before looking. Then run the lie detector on the result: at $x = 1$, do the original and the simplified version agree?

The one thing to remember

A variable is a number-in-waiting, and an expression is a machine that becomes a number the moment you feed it one. Only matching kinds count together; an outside factor — sign included — reaches every term inside; and when you’re not sure a simplification is legal, substitute a small number and let arithmetic be the judge.

The language of algebra

A variable (like $x$) is a placeholder for a number. A term is a single piece: $3x$ or $-7$. The number in front of a variable is its coefficient (in $3x$, the $3$). A plain number is a constant.

Like terms

Like terms have the exact same variable part. $3x$ and $5x$ are like terms; $3x$ and $5$ are not. You can only add or subtract like terms — count the $x$‘s together, count the plain numbers together.

$3x + 2 + 5x - 6 = 8x - 4$

Evaluating

To evaluate, replace the variable with a number and compute. If $x = 4$: $3x + 2 = 3(4) + 2 = 14$.

The distributive property

A number outside parentheses multiplies every term inside: $a(bx + c) = abx + ac$.

$5(2x - 3) = 5 \cdot 2x - 5 \cdot 3 = 10x - 15$

Putting both together, the signature example:

Quick reference