Introduction to Expressions

Q: Does the sign in front of a term belong to it?

Yes. In $8x - 5$, the second term is $-5$, not $5$. Keeping the sign attached to its term is the habit that prevents most sign errors when you start combining like terms.

An expression is a group of numbers and variables joined by addition, subtraction, multiplication, or division — something like $8x - 5 + 3y - 12$ . There is no equal sign, so there is nothing to solve. Expressions are the raw material of algebra; equations come later.

Before you can simplify or evaluate anything, you need four words: term, variable, coefficient, and constant. Every instruction you will hear this year — combine the like terms, divide by the coefficient, move the constant — uses this vocabulary, so it is worth locking in now.

The four vocabulary words

A term is each piece of the expression separated by a $+$ or $-$ sign. In $8x - 5 + 3y - 12$ the terms are $8x$ , $-5$ , $3y$ , and $-12$ — four terms in total. Notice the sign in front of a term belongs to that term.

A variable is a letter standing in for an unknown number. This expression has two variables, $x$ and $y$ .

A coefficient is the number multiplied by a variable. The coefficient of $x$ is $8$ , and the coefficient of $y$ is $3$ .

A constant is a term with no variable attached — just a plain number. Here the constants are $-5$ and $-12$ .

The invisible coefficient

When a variable appears with no number in front, its coefficient is $1$ . The expression $x - 7$ really means $1x - 7$ , so the coefficient of $x$ is $1$ . Likewise $-m$ means $-1 \cdot m$ , so its coefficient is $-1$ — the negative sign counts.

This matters constantly later on. When you combine like terms or divide both sides of an equation, that invisible $1$ has to be treated like any other number.

Worked examples

Example 1: label every part

Identify the terms, variables, coefficients, and constants in $8x - 5 + 3y - 12$ .

Split at each

+

-

sign

8x,\; -5,\; 3y,\; -12

Count the terms

4 \text{ terms}

Find the letters

x \text{ and } y

Numbers attached to variables

8 \text{ and } 3

Terms with no variable

-5 \text{ and } -12

Answer: $4$ terms; variables $x$ and $y$ ; coefficients $8$ and $3$ ; constants $-5$ and $-12$

Example 2: the invisible coefficient

Identify the parts of $x - 7$ .

Split into terms

x,\; -7

Rewrite the first term with its hidden

1

1x

The plain number is the constant

-7

Answer: $2$ terms; coefficient of $x$ is $1$ ; constant is $-7$

Example 3: a negative coefficient

What is the coefficient of $m$ in $-m + 5n - 2$ ?

Rewrite

-m

explicitly

-1 \cdot m

Read off the number attached to

m

-1

Answer: $-1$

Try one yourself

Common questions

What is the difference between an expression and an equation?

An equation has an equal sign; an expression does not. You solve an equation, but you can only simplify or evaluate an expression. $3x + 5$ is an expression; $3x + 5 = 20$ is an equation.

Does the sign in front of a term belong to it?

Yes. In $8x - 5$ , the second term is $-5$ , not $5$ . Keeping the sign attached to its term is the habit that prevents most sign errors when you start combining like terms.

Is a constant also a term?

Yes. A constant is simply a term that happens to have no variable. In $3y - 7 + 2y$ there are three terms, and one of them — $-7$ — is a constant.

Can a coefficient be a fraction or a decimal?

Yes. In $\dfrac{1}{2}x + 4$ the coefficient of $x$ is $\dfrac{1}{2}$ , and in $0.75n$ the coefficient is $0.75$ . A coefficient is any number multiplied by a variable.

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Introduction to Expressions

The four vocabulary words

The invisible coefficient

Worked examples

Example 1: label every part

Example 2: the invisible coefficient

Example 3: a negative coefficient

Try one yourself

Common questions

Keep learning

Numerical Expressions

Combining Like Terms

Distributive Property

Properties of Real Numbers