coefficient
A coefficient is the number stuck to the front of a variable, telling you how many copies of it you have. In “3x” the coefficient is 3, which says “three x’s.” Just as 3 apples means apple + apple + apple, 3x means x + x + x.
More carefully, the coefficient is the numerical factor that multiplies the variable part of a term. In “−5y” it is −5; in “x^2/4” it is 1/4; and in a bare “x” the coefficient is the invisible 1, since x means 1·x. Coefficients carry their sign, so the term “−7t” has coefficient −7, not 7.
Coefficients matter because they steer the behaviour of an expression: in 3x the 3 controls the steepness of the line, and in ax^2 + bx + c the numbers a, b, c shape the whole parabola. Identifying coefficients correctly is the first step in combining like terms, factoring, and reading off what an equation will do.
In 4x − x + 2x, the coefficients are 4, −1, and 2; adding them gives 5x.
Combining like terms is really just adding their coefficients.
The coefficient of a lone variable is 1, and of its negative is −1: x has coefficient 1, and −x has coefficient −1. Forgetting this invisible 1 is a common slip.