How do coefficients, variables, and constants work in an expression?

First, separate the expression into its terms: 4x and 7. A term may be a constant or a product of a constant and variables; the multiplying constant is its coefficient. [S1]

In 4x + 7:

1. x is the variable—the letter representing a quantity.
2. 4 is the coefficient of x; it multiplies the variable.
3. 7 is the constant term because it stands by itself, with no variable attached.

A quick way to check is to ask what is being multiplied by the variable in each variable-containing term, then look for any number term left on its own. The + sign separates the terms here. An expression can combine numbers and variables with operations. [S1]

The coefficient may be implied rather than written. OpenStax gives x as an example with implied coefficient 1. [S1] So x means 1x, and the coefficient is 1. In -y, the minus sign means the coefficient is negative: -1, since -y can be read as (-1)y.

For comparison, in 3x - y + 8, the terms are 3x, -y, and 8. The coefficients are 3 and -1; the variables are x and y; and 8 is the constant term. Keep the sign attached to its term when identifying the coefficient. The 8 is positive because it is added.

A variable is a symbol for a quantity; the letter itself does not inherently name one fixed number in every context. [S1] In a particular problem, context or other information may specify what value it represents. If no value is given, you can still identify its role without guessing a value.

For 2xy + 5x - 3, read the terms as 2xy, 5x, and -3:

1. In 2xy, the coefficient is 2, and the variable part is xy (both x and y appear).
2. In 5x, the coefficient is 5, and the variable is x.
3. -3 is the constant term.

The coefficient multiplies the variable part of its own term. It is not necessarily the number multiplying just one letter when a term contains more than one variable.

Use this annotated example: -2a + b + 9.

• -2a: variable a; coefficient -2.
• b: variable b; implied coefficient 1.
• 9: constant term; it has no variable attached.

Terminology reference: a variable is a letter representing a quantity; a coefficient is the multiplying constant in a variable-containing term; a constant term is a term that is just a constant. An expression can be a number, a variable, or a combination of numbers and variables joined by operations. [S1]

One useful caution: don’t call every number in an expression a constant term. In -2a, the number -2 is a coefficient, not a separate standalone term. These labels describe the number’s role in that expression; they do not tell you the variable’s value. The definitions here help identify parts, but an expression alone may not provide a value for its variables.