Variable

Start with the idea

Suppose a sealed envelope contains some counters. You can discuss the number inside without opening it: call that number $n$.

The letter is simply a label. It lets us describe a number we do not yet know—or a number that can change.

Three common jobs

An unknown to find

In $x+4=11$, only one value makes the equation true: $x=7$.

A quantity that changes

In $d=60t$, the time $t$ can vary and the distance $d$ changes with it. If $t=2$, then $d=120$; if $t=3.5$, then $d=210$.

A number chosen from a set

In “$n$ is an even integer,” $n$ could be $-4$, $0$, $18$, or any other even integer. It is not waiting for one hidden answer.

Reading algebra

Letters beside numbers mean multiplication:

The number multiplying the variable is the Coefficient. In $3n+2$, the coefficient is $3$, the variable is $n$, and the constant term is $2$.

Solving for a variable

An equation is balanced when both sides have the same value. Solving means finding the value that keeps that balance.

The real action is not “move the 4.” It is subtracting the same amount from both sides.

Variables in formulas

In $A=lw$, the variables represent rectangle area, length, and width. If $l=8$ cm and $w=3$ cm, then

Units matter because the variables represent quantities, not merely marks on a page.

Common misunderstandings