value of a variable

The value of a variable is the actual number it stands for at a given moment. A variable is like an empty box with a name; its value is what you have put inside the box. When you say “let x = 5,” you are assigning the value 5 to the variable x for the calculation that follows.

Knowing a variable’s value is what turns a general expression into a definite number. Until x has a value, 3x + 1 is just a recipe; the instant x = 4, the recipe yields the single number 13. Supplying values for variables and computing the result is exactly what evaluation and substitution are for.

Two related ideas hide here. A variable may be allowed to take many values in turn (as the input to a function does), so it has a range of possible values rather than just one. In an equation, by contrast, the values of interest are precisely those that make the statement true — the solutions. Either way, “the value” means the specific number standing in for the variable in the situation you are considering.