Variables and Algebraic Expressions — Writing, Evaluating, and Simplifying

By MathPal Editorial Team·Published Apr 17, 2026

Grade: 6-7 | Topic: Algebra

What You Will Learn

In this lesson you will understand what variables are and why we use them, learn how to translate everyday English phrases into algebraic expressions, practice evaluating expressions by substituting numbers, and simplify expressions by combining like terms.

Theory

What is a Variable?

A variable is a letter that stands for a number we do not know yet (or a number that can change). The most common variable letters are $x$, $y$, and $n$, but any letter can be used.

Think of a variable as a box with a label: the label is the letter, and any number can go inside the box. When we write $x + 5$, we mean "some number plus five." If $x = 3$, the result is $8$. If $x = 10$, the result is $15$.

What is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables, and operations (addition, subtraction, multiplication, division). Here are some examples:

• $2x + 7$
• $5n - 3$
• $4a + 2b$
• $\frac{y}{3} + 9$

Notice that expressions do not have an equals sign. Once you add an equals sign (like $2x + 7 = 15$), it becomes an equation.

Parts of an Expression

Term	Meaning	Example in $3x + 5$
Term	A single piece separated by + or -	$3x$ and $5$
Coefficient	The number multiplied by a variable	$3$ (in $3x$)
Constant	A term with no variable	$5$
Variable	The letter	$x$

Writing Expressions from Words

Algebra often starts by translating words into symbols. Here is a reference for common phrases:

English phrase	Operation	Algebraic form
a number plus 8	addition	$n + 8$
5 more than $x$	addition	$x + 5$
a number minus 3	subtraction	$n - 3$
7 less than $y$	subtraction	$y - 7$ (not $7 - y$)
twice a number	multiplication	$2n$
the product of 4 and $x$	multiplication	$4x$
a number divided by 6	division	$\frac{n}{6}$
half of $z$	division	$\frac{z}{2}$

Watch out for "less than" — the phrase "7 less than $y$" means $y - 7$, not $7 - y$. The number after "less than" comes first in the expression.

Evaluating Expressions

To evaluate an expression means to substitute a given value for each variable and then compute the result.

For $3x + 2$ when $x = 4$:

$3(4) + 2 = 12 + 2 = 14$

Always use parentheses when substituting to avoid sign errors, especially with negative numbers.

Simplifying by Combining Like Terms

Like terms are terms that have exactly the same variable part. You simplify by adding or subtracting their coefficients.

• $5x + 3x = 8x$ (same variable $x$, add coefficients)
• $7y - 2y = 5y$
• $4x + 3 + 2x + 9 = 6x + 12$ (combine $x$-terms and constants separately)

Terms like $3x$ and $3y$ are not like terms because the variables differ. Similarly, $x$ and $x^2$ are not like terms because the exponents differ.

Worked Examples

Example 1: Translate words to algebra

Write an algebraic expression for "three times a number, decreased by 11."

Let the number be $n$.

"Three times a number" is $3n$. "Decreased by 11" means subtract 11.

$3n - 11$

Example 2: Evaluate an expression

Evaluate $2a - 3b + 10$ when $a = 5$ and $b = -2$.

Substitute:

$2(5) - 3(-2) + 10$

$= 10 - (-6) + 10$

$= 10 + 6 + 10$

$= 26$

Notice that subtracting a negative number becomes addition: $-3(-2) = +6$.

Example 3: Simplify an expression

Simplify $8x + 4 - 3x + 7 - x$.

Group the $x$-terms: $8x - 3x - x = 4x$.

Group the constants: $4 + 7 = 11$.

$8x + 4 - 3x + 7 - x = 4x + 11$

Example 4: Multi-variable expression

Simplify $5a + 2b - 3a + 4b + 1$.

Group $a$-terms: $5a - 3a = 2a$.

Group $b$-terms: $2b + 4b = 6b$.

Constants: $1$.

$5a + 2b - 3a + 4b + 1 = 2a + 6b + 1$

Common Mistakes

Mistake 1: Translating "less than" backwards

❌ "5 less than $n$" $\rightarrow$ $5 - n$

✅ "5 less than $n$" means start with $n$ and take away 5: $n - 5$. The phrase "less than" reverses the order.

Mistake 2: Combining unlike terms

❌ $3x + 4y = 7xy$

✅ You can only combine terms with the same variable. Since $3x$ and $4y$ have different variables, the expression $3x + 4y$ is already simplified. And $xy$ is a completely different term (it means $x$ times $y$).

Mistake 3: Forgetting the sign when substituting negatives

❌ Evaluating $x^2$ when $x = -3$: "$x^2 = -3^2 = -9$"

✅ Use parentheses: $(-3)^2 = 9$. Without parentheses, $-3^2$ means $-(3^2) = -9$, which is a different result. Always wrap negative substitutions in parentheses.

Practice Problems

1. Write an algebraic expression: "the sum of a number and 12, divided by 4."

Show Answer

$\frac{n + 12}{4}$

"The sum of a number and 12" is $n + 12$. "Divided by 4" means the whole sum is divided.

2. Evaluate $4x - 7$ when $x = 3$.

Show Answer

$4(3) - 7 = 12 - 7 = 5$

3. Evaluate $a^2 + 2a - 1$ when $a = -2$.

Show Answer

$(-2)^2 + 2(-2) - 1 = 4 + (-4) - 1 = -1$

4. Simplify: $6m + 9 - 2m + 3m - 4$.

Show Answer

Group $m$-terms: $6m - 2m + 3m = 7m$.

Group constants: $9 - 4 = 5$.

Simplified: $7m + 5$.

5. Simplify: $3x + 5y - x + 2y - 8$.

Show Answer

Group $x$-terms: $3x - x = 2x$.

Group $y$-terms: $5y + 2y = 7y$.

Constants: $-8$.

Simplified: $2x + 7y - 8$.

Summary

• A variable is a letter representing an unknown or changeable number.
• An algebraic expression combines numbers, variables, and operations — no equals sign.
• To evaluate, substitute numbers for variables and compute.
• To simplify, combine like terms (terms with the same variable and exponent).
• Careful translation of English phrases into algebra is a foundational skill for solving equations.

Related Topics

• Linear Equations — the next step: set expressions equal to a value and solve for the variable.
• Solving One-Step Equations — practice solving simple equations using inverse operations.
• Order of Operations — make sure you evaluate expressions in the correct order (PEMDAS).

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