Key takeaway: A coefficient is the number that multiplies a variable in an algebraic term, such as 9 in 9a, and when no number is written the coefficient is 1.
Grade 66.EE.A.2bExpressions and Equations
Definition of Coefficient
A coefficient is the numerical factor sitting in front of a variable. In the term \(9a\), the 9 says there are nine copies of \(a\). In \(-4b\), the multiplier is \(-4\), and the sign belongs to it.
If a letter stands alone, its multiplier is hidden. The term \(a\) means \(1a\), and \(-b\) means \(-1b\). A term with no letter at all, like 6, is called a constant. These multipliers show how fast an expression grows when the variable increases: in \(9a\), every extra unit of \(a\) adds 9 to the total.
Formula and notation
In a term written as a product,
\[9a = 9 \times a\]
the number 9 comes first and the letter follows. Fractions and decimals work too, as in \(\frac{2}{3}x\) or \(0.5y\). When two letters are multiplied, like \(7xy\), the numerical factor is still 7.
Key facts
- The sign in front of a term belongs to its numerical factor.
- A bare letter has an unwritten 1; a letter with a minus sign has \(-1\).
- Adding like terms means adding these multipliers and keeping the variable part.
- In a formula such as \(d = 55t\), the number 55 is the speed.
Coefficient example
Name the number attached to each letter in \(9a – 4b + a + 6\), then give the coefficient of \(a\) after the two \(a\)-terms are combined.
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\(9a\) has the multiplier 9.
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\(-4b\) has \(-4\), including its sign.
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The lone \(a\) has an unwritten 1. The number 6 is a constant.
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Combine the \(a\)-terms: \(9 + 1 = 10\), so that part is \(10a\).
The multipliers are 9, \(-4\) and 1, and after combining, the coefficient of \(a\) is 10.
Common mistake
Watch out
Priya says the number in front of the letter in \(-y\) is 1 because “there is no number.”
Why it is wrong
A missing number means 1, but the minus sign is still there. The term \(-y\) equals \(-1 \times y\), so it carries \(-1\). Ignoring the sign causes errors when terms are combined.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Expressions and Equations