Variables and Algebraic Expressions

Up to this point, every math problem you have worked with contained only specific numbers. Algebra introduces a powerful new idea: using letters to stand for numbers we do not know yet. These letters are called variables, and they let us write general rules and solve problems where information is missing.

What Is a Variable?

A variable is a letter — most often $x$, $y$, or $n$ — that represents a number whose value can change or is not yet known. Think of it as a blank space in a sentence that you plan to fill in later.

For example, if a store charges $8 per item, the total cost for $n$ items is:

$\text{Total cost} = 8 \times n$

We do not know how many items the customer will buy, so $n$ is a variable. Once the customer brings 5 items to the counter, we substitute $n = 5$ and get $8 \times 5 = 40$.

Algebraic Expressions vs. Numeric Expressions

A numeric expression contains only numbers and operations:

$3 + 7 \times 2$

An algebraic expression contains at least one variable:

$3 + 7x$

Both follow the same order of operations rules, but an algebraic expression cannot be reduced to a single number until every variable has a known value.

The Vocabulary: Terms, Coefficients, and Constants

Every algebraic expression is built from terms joined by addition or subtraction.

Consider the expression:

$5x + 3y - 9$

Coefficients

A coefficient is the number multiplied by a variable. If no number is written, the coefficient is $1$:

$x = 1 \cdot x$

Similarly, $-y$ has a coefficient of $-1$.

Constants

A constant is a term with no variable. Constants keep the same value no matter what the variables equal.

Example 1: Identify the parts of $4a - 7b + 12$

There are three terms. Two have variables; one is a constant.

Example 2: Identify the parts of $-m + 6$

The coefficient of $m$ is $-1$, not $1$.

Writing Expressions from Word Descriptions

One of the most practical skills in algebra is turning a word description into a mathematical expression.

Example 3: “Five more than a number”

Let $n$ represent the unknown number.

$n + 5$

Example 4: “Three times a number, decreased by two”

$3n - 2$

Example 5: “The quotient of a number and four, increased by seven”

$\frac{n}{4} + 7$

Notice that the order matters. “Five more than a number” gives $n + 5$, while “five less than a number” gives $n - 5$.

Expression vs. Equation

An expression is a phrase — it has no equals sign:

$2x + 7$

An equation is a complete sentence — it says two things are equal:

$2x + 7 = 15$

Expressions are simplified. Equations are solved. This distinction matters throughout all of algebra.

Multiplication Conventions in Algebra

Algebra uses shorthand to keep things clean:

The multiplication sign ($\times$) is almost never written between a number and a variable or between two variables, because it could be confused with the letter $x$.

Real-World Application: Retail — Calculating a Discounted Total

A retail associate needs to calculate the final price for a customer buying $n$ shirts at $24 each with a $10 coupon applied to the total. The algebraic expression for the total is:

$24n - 10$

If the customer buys $n = 3$ shirts:

$24(3) - 10 = 72 - 10 = 62$

The total is $62. Notice how the expression lets the associate handle any number of shirts — just substitute the new value for $n$.

Real-World Application: Nursing — IV Drip Rate Expression

A nurse uses the formula for IV flow rate:

$\text{Rate} = \frac{V \times d}{t}$

where $V$ is the volume in mL, $d$ is the drop factor (drops per mL), and $t$ is the time in minutes. This is an algebraic expression with three variables. Once the specific prescription values are known, the nurse substitutes them to find the drip rate in drops per minute.

Common Mistakes to Avoid

1. Forgetting the invisible coefficient of 1. The expression $x$ means $1 \cdot x$, and $-x$ means $-1 \cdot x$. When combining like terms later, this distinction is essential.

2. Confusing terms and factors. Terms are separated by $+$ or $-$ signs. Factors are things being multiplied within a single term. In $3xy$, the term is $3xy$, and its factors are $3$, $x$, and $y$.

3. Misreading subtraction order. “Ten less than a number” is $n - 10$, not $10 - n$. The phrase “less than” reverses the order compared to how you read it.

4. Assuming an expression equals something. An expression like $4x + 1$ does not equal a specific value. It is not an equation until you write an equals sign and a second side.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A variable is a letter that represents an unknown or changing number
• An algebraic expression contains at least one variable, while a numeric expression has only numbers
• Terms are the building blocks of expressions, separated by addition or subtraction
• The coefficient is the number multiplying a variable; a constant has no variable
• An expression has no equals sign — it is simplified, not solved
• An equation has an equals sign — it states that two expressions are equal
• Converting word descriptions into algebraic expressions is a foundational skill for all of algebra

Return to Pre-Algebra for more topics in this section.