Algebraic Expressions

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An algebraic expression is a math phrase made of numbers, variables and operations, like 2x + 7 or 5ab - 3. It has no equals sign. To evaluate an algebraic expression, replace each variable with its value and follow the order of operations. To write one, turn the words of a problem into symbols.

Before this lesson: You should know what variables are and how to use the order of operations.

What Is an Algebraic Expression?

An algebraic expression combines numbers, variables and operations such as adding, subtracting, multiplying and dividing. Here are a few examples:

  • 2x + 7
  • 5ab - 3
  • \frac{n}{4}
  • x^2 - 9

An expression is like a phrase. It names a value, but it doesn’t make a full statement. An equation is like a sentence. It says that two things are equal.

Equals sign?ExampleWhat you do with it
ExpressionNo2x + 7Evaluate it or simplify it
EquationYes2x + 7 = 15Solve it to find x

Terms of an Expression

The terms of an expression are the parts that are added or subtracted.

The expression 5x − 2y + 8 split into three terms: 5x, −2y and 8.
5x − 2y + 8 has three terms: 5x, −2y and 8.

Each term keeps the sign in front of it. So the second term of 5x - 2y + 8 is -2y, not 2y.

Inside a term, everything is multiplied. That means 6ab is one term, not three.

Terms with the same variable part are called like terms, such as 3x and -7x. You’ll learn to put them together in combining like terms.

How to Write Algebraic Expressions

Word problems use key words for each operation. This table shows the most common ones.

OperationKey wordsIn wordsExpression
Addsum, plus, more than, increased by, total5 more than a numbern + 5
Subtractdifference, minus, less than, decreased by4 less than a numbern - 4
Multiplyproduct, times, twice, oftwice a number2n
Dividequotient, divided by, perthe quotient of a number and 3\frac{n}{3}

Watch out for “less than” and “subtracted from.” They flip the order. “4 less than a number” means start with the number and take away 4, so it is n - 4, not 4 - n.

Use parentheses to keep a group together. “Three times the sum of a number and 2” is 3(n + 2). You can multiply out parentheses like these with the distributive property.

How to Evaluate an Algebraic Expression

To evaluate means to find the value of an expression. Follow these steps:

  1. Write the expression.
  2. Replace each variable with its value. Put each value in parentheses.
  3. Simplify with the order of operations: parentheses, exponents, multiply and divide, then add and subtract.

Worked Examples

Example 1: Write an expression from words

Write an expression for “seven less than three times a number.”

Step 1. Let n = the number.

Step 2. “Three times a number” is 3n.

Step 3. “Seven less than” means take 7 away from 3n.

Answer: 3n - 7

Example 2: Use parentheses

Write an expression for “half the sum of a number and 10.”

Step 1. Let n = the number. The sum of the number and 10 is n + 10.

Step 2. Half of something means divide it by 2. Divide the whole sum, not just the 10.

Answer: \frac{n + 10}{2}. You could also write \frac{1}{2}(n + 10).

Example 3: Evaluate with two variables

Evaluate 2a + 3b when a = 4 and b = -5.

Step 1. Replace each variable with its value in parentheses.

Step 2. Multiply, then add.

\begin{aligned}
2a + 3b &= 2(4) + 3(-5) \\
&= 8 + (-15) \\
&= -7
\end{aligned}

Answer: -7

Example 4: Evaluate with an exponent and a fraction

Evaluate \frac{x^2 - 1}{y} + xy when x = -3 and y = 2.

Step 1. Replace each variable with its value in parentheses.

Step 2. Work out the top of the fraction first, because the fraction bar groups it.

Step 3. Divide and multiply, then add.

\begin{aligned}
\frac{x^2 - 1}{y} + xy &= \frac{(-3)^2 - 1}{2} + (-3)(2) \\
&= \frac{9 - 1}{2} + (-6) \\
&= 4 + (-6) \\
&= -2
\end{aligned}

Answer: -2

Example 5: Word problem

A movie ticket costs $11 and a bag of popcorn costs $6. Write an expression for the cost of t tickets and p bags of popcorn. Then find the cost of 4 tickets and 2 bags of popcorn.

Step 1. The tickets cost 11t dollars, and the popcorn costs 6p dollars.

Step 2. The total is the sum.

\text{cost} = 11t + 6p

Step 3. Replace t with 4 and p with 2.

11(4) + 6(2) = 44 + 12 = 56

Answer: The cost is 11t + 6p dollars. For 4 tickets and 2 bags of popcorn, it is $56.

Common Mistakes

MistakeWhat it looks likeHow to fix it
Flipping “less than”“5 less than x” written as 5 - xStart with x and take away 5: x - 5.
Leaving out parentheses“Twice the sum of x and 3″ written as 2x + 3Double the whole sum: 2(x + 3).
Squaring a negative without parenthesesIf x = -4, then x^2 = -16(-4)^2 = 16
Adding before multiplying3 + 2x at x = 5 becomes 5 \cdot 5 = 25Multiply first: 3 + 2(5) = 3 + 10 = 13.
Trying to “solve” an expression“Solve 4x + 1“An expression has no equals sign. You evaluate or simplify it. Only equations are solved.

Quick check: Read your expression back in words. It should say the same thing as the problem.

Writing an Expression from a Pattern

Sometimes the numbers come in a table. Look at how the output changes each time the input goes up by 1.

nCost
113
221
329
437

Each time n goes up by 1, the cost goes up by 8. So the expression has 8n. When n = 1, 8n is 8, but the cost is 13. That is 5 more, so the expression is 8n + 5.

Check another row: when n = 3, 8(3) + 5 = 29. It matches.

Practice Problems

Write or evaluate each expression. Show your steps.

  1. How many terms are in 6x^2 - x + 4y - 10? List them.
  2. Write an expression for “the product of 9 and a number.”
  3. Write an expression for “12 less than a number.”
  4. Write an expression for “the sum of a number and 6, divided by 5.”
  5. Write an expression for “8 more than twice a number.”
  6. Evaluate 5x - 2y when x = 3 and y = 4.
  7. Evaluate a^2 - 3a + 2 when a = -2.
  8. Evaluate \frac{m + n}{m - n} when m = 7 and n = 3.
  9. A taxi charges $3.50 plus $2 for each mile. Write an expression for the cost of m miles. Then find the cost of a 12-mile ride.
  10. Write an expression for the pattern: when n is 1, 2, 3 and 4, the value is 7, 11, 15 and 19.

Answers

Show answers
  1. 4 terms: 6x^2, -x, 4y and -10.
  2. 9n
  3. n - 12
  4. \frac{n + 6}{5}
  5. 2n + 8
  6. 5(3) - 2(4) = 15 - 8 = 7
  7. (-2)^2 - 3(-2) + 2 = 4 + 6 + 2 = 12
  8. \frac{7 + 3}{7 - 3} = \frac{10}{4} = \frac{5}{2}
  9. The cost is 2m + 3.50 dollars. For 12 miles: 2(12) + 3.50 = 24 + 3.50 = 27.50, so the ride costs $27.50.
  10. The value goes up by 4 each time, so start with 4n. When n = 1, 4n = 4, and 7 is 3 more. The expression is 4n + 3.

Algebraic Expressions FAQ

What is an algebraic expression?

It is a combination of numbers, variables and operations with no equals sign, such as 4x - 9. It stands for a value that depends on the variables.

What is the difference between an expression and an equation?

An equation has an equals sign and an expression does not. You solve an equation like 4x - 9 = 15, but you evaluate or simplify an expression like 4x - 9.

How do you evaluate an algebraic expression?

Replace each variable with its value, using parentheses. Then simplify with the order of operations.

What are the parts of an algebraic expression?

An expression is made of terms. Each term can have a coefficient and variables, or be a constant. In 3x + 5, 3x and 5 are terms, 3 is the coefficient and 5 is the constant.

Summary

  • An algebraic expression is made of numbers, variables and operations, with no equals sign.
  • Terms are the parts that are added or subtracted. Each term keeps its sign.
  • Key words tell you the operation. “Less than” flips the order: “4 less than n” is n - 4.
  • Use parentheses to keep a group together.
  • To evaluate, substitute each value in parentheses and follow the order of operations.

Previous lesson: Variables

Next lesson: Combining Like Terms

Back to the Algebra 1 Index

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