The Lesson

A coefficient is a number that is placed in front of a variable. The coefficient is multiplying the variable.

Dictionary Definition

The Oxford English Dictionary defines a coefficient as "a number or quantity placed (usually) before and multiplying another quantity known or unknown."

Real Examples of Coefficients

It is easier to understand coefficients with examples.
  • Consider the term below:

    6 x 6 is the coefficient of x.
  • Consider the expression below. One of the coefficients is a letter rather than a number:

    a x squared plus x plus 2 y. a (a constant) is the coefficient of x2. 1 is the coefficient of x, even though no number or letter is written in front of it (see Note). 2 is the coefficient of y.

Understanding Coefficients

Think of a letter in algebra as an object. We can count the objects. x is one x. If we add one x to one x, we get two x's.
x + x = 2x
The coefficient of 2 is telling us we have 2 x's. Another way of saying this is that the 2 is multiplying the x after it:
2x = 2 × x = x + x
If there was a coefficient of 3 in front of an x, we would have 3 x's:
3x = 3 × x = x + x + x

Letters As Coefficients

Letters can be used as coefficients. For example, a standard linear equation is y = mx + c. The m is the coefficient of the x term.The letter still represents a number. The m could be a 2 or a −3 for different linear equations.

Lesson Slides

The slider below has some more real examples of coefficients.

A Term Without a Number in Front of it Has a Coefficient of 1

Sometimes a term will not have a number in front of it.

a variable without a coefficient. Don't be fooled. It still has a coefficient. The coefficient is 1.

a variable without a coefficient written in front of it has a coefficient of 1.

Positive, Negative and Fractional Coefficients

Coefficients can be positive, negative and fractional.

a positive coefficient, a negative coefficient and a fractional coefficient