Variables & Algebraic Expressions

Q: Is the coefficient of $x$ in $7 - 2x$ equal to $2$ or $-2$?

It's $-2$. The sign in front of a term belongs to that term, so the terms of $7 - 2x$ are $7$ and $-2x$. Dropping that minus sign is the single most common mistake with this skill.

Q: Can a constant be negative?

Yes. In $6y - 11$, the constant is $-11$, because the minus sign stays with its term. A constant is any term with no variable, whatever its sign.

In arithmetic, every problem is made of numbers you can see. Algebra adds one new character: the variable — a letter like $x$ or $n$ that stands for a number we don't know yet. An algebraic expression is a phrase built from variables, numbers, and operations, like $4x + 7$ . Notice what it doesn't have: an equals sign. An expression describes a quantity; an equation states that two quantities are equal.

Before you can simplify or solve anything, you need to name the parts of an expression: its terms, its coefficients, and its constant. Every skill that follows — combining like terms, distributing, factoring — is described in those three words, so it pays to lock in the vocabulary now.

Terms: the building blocks

Terms are the pieces of an expression separated by $+$ and $-$ signs. The expression $4x + 7$ has two terms: $4x$ and $7$ . The expression $5x - 3y + 8$ has three terms: $5x$ , $-3y$ , and $8$ .

The sign sitting in front of a term belongs to that term. In $5x - 3y + 8$ , the middle term is $-3y$ , not $3y$ . Reading every term together with its sign is a small habit that prevents most sign errors later in algebra.

Coefficients and constants

The coefficient is the number multiplied by the variable. In $4x + 7$ , the coefficient of $x$ is $4$ . In $10 - 3y + 6x$ , the coefficient of $y$ is $-3$ — the minus sign travels with its term.

The constant is the term with no variable: a plain number whose value never changes. In $4x + 7$ the constant is $7$ . In $6y - 11 + 2x$ the constant is $-11$ , because the minus sign stays attached.

A variable written alone has an invisible coefficient of $1$ : the term $x$ means $1x$ , and $-y$ means $-1y$ . Nobody bothers to write the $1$ , but it is always there.

What the parts mean in a real situation

The parts usually stand for something concrete. A gym that charges 20 dollars to join plus 8 dollars each month costs $8m + 20$ dollars after $m$ months. The variable $m$ counts the months. The coefficient $8$ is the amount that repeats with every month. The constant $20$ is the one-time fee that never changes, no matter how long you stay. When you can read an expression this way, the algebra stops being abstract.

Worked examples

Example 1: count the terms

How many terms are in $3a + 5b - 2$ ?

Terms are separated by

+

and

-

signs

3a + 5b - 2

List each term with its sign

3a,\ 5b,\ -2

Answer: $3$ terms

Example 2: name every part

In $5x - 3y + 8$ , find the number of terms, the coefficient of $y$ , and the constant.

Split the expression at the signs

5x,\ -3y,\ 8

Count the terms

3

The term with

y

-3y

, so its coefficient is

-3

The term with no variable is the constant

8

Answer: $3$ terms; coefficient of $y$ is $-3$ ; constant is $8$

Example 3: the invisible coefficient

What is the coefficient of $x$ in $x - 4y + 7$ ?

The term with

x

is written alone

x

A lone variable means

1

times that variable

x = 1x

Answer: $1$

Try one yourself

Common questions

What's the difference between an expression and an equation?

An equals sign. $4x + 7$ is an expression — it names a quantity. $4x + 7 = 19$ is an equation — it claims two quantities are equal, and it can be solved. You simplify or evaluate expressions; you solve equations.

Is the coefficient of

x

7 - 2x

equal to

2

-2

It's $-2$ . The sign in front of a term belongs to that term, so the terms of $7 - 2x$ are $7$ and $-2x$ . Dropping that minus sign is the single most common mistake with this skill.

Can a constant be negative?

Yes. In $6y - 11$ , the constant is $-11$ , because the minus sign stays with its term. A constant is any term with no variable, whatever its sign.

Why is a letter used instead of a number?

Because the number isn't known yet, or because it changes. In the gym expression $8m + 20$ , the months $m$ could be $1$ , $6$ , or $24$ — the variable lets one expression cover every case at once.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Variables & Algebraic Expressions

Terms: the building blocks

Coefficients and constants

What the parts mean in a real situation

Worked examples

Example 1: count the terms

Example 2: name every part

Example 3: the invisible coefficient

Try one yourself

Common questions

Keep learning

Writing Expressions from Words

Evaluating Expressions

Combining Like Terms

The Distributive Property