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Variable, Constant, Unknown

Three words that look interchangeable but are not. A variable can roam, a constant is pinned down, and an unknown is a fixed-but-hidden number we are hunting. Telling them apart is half of reading algebra.

Same letters, different jobs

A variable is a letter that is *allowed to change* — it ranges over many possible numbers, and we want to see how something depends on it. In y = 2x, the x is a variable: feed in any number and y responds. A constant is a quantity that *stays fixed* throughout a problem; the 2 here never moves. An unknown is special: it is a *single fixed number we do not yet know* but intend to find, like the x in 2x + 1 = 7, which can only be 3.

Coefficients, constant terms, and a fourth word

When a number multiplies a variable, it is the variable's coefficient. In 2x the coefficient is 2. A number standing alone, multiplied by no variable, is a constant term — the +1 in 2x + 1. There is also a subtler word, parameter: a letter held fixed *for now*, defining a whole family of related cases, which we may later change to get a different case.

Look at:   y = m·x + b

  x  is the VARIABLE   (it roams; y depends on it)
  m  is a PARAMETER    (fixed per line, but choose it
                        differently for a different line)
  b  is a PARAMETER too (the line's height at x = 0)
  y  is the OUTPUT      (another variable, set by x)

Now pin m = 2 and b = 1:   y = 2x + 1
  the 2 is the COEFFICIENT of x
  the 1 is the CONSTANT TERM
The line formula shows all four roles at once.

When you finally pick an actual number for a variable, that number is the value of the variable. Choosing x = 5 and finding y is the act of substitution, which we look at closely in guide 4. For now the key habit is simply to *ask of every letter*: is it free to vary, is it pinned as a constant, or is it a hidden number I am trying to uncover?