Algebraic Expressions: Write and Evaluate

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.

Extra practice: The Algebra 1 Unit 1 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

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?	Example	What you do with it
Expression	No	$2x + 7$	Evaluate it or simplify it
Equation	Yes	$2x + 7 = 15$	Solve 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.

Operation	Key words	In words	Expression
Add	sum, plus, more than, increased by, total	5 more than a number	$n + 5$
Subtract	difference, minus, less than, decreased by	4 less than a number	$n - 4$
Multiply	product, times, twice, of	twice a number	$2n$
Divide	quotient, divided by, per	the 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:

Write the expression.
Replace each variable with its value. Put each value in parentheses.
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. Choose a variable. Let $n$ = the number.

3n

Step 2. Translate “three times a number.”

3n - 7

Step 3. “Seven less than” means take 7 away from it. Write the 7 last.

Answer: $3n - 7$

Example 2: Use parentheses

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

Step 1. Choose a variable. Let $n$ = the number.

n + 10

Step 2. Write the sum of the number and 10.

\frac{n + 10}{2}

Step 3. 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 $b = -5$ .

2a + 3b = 2(4) + 3(-5)

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

2(4) + 3(-5) = 8 + 3(-5)

Step 2. Multiply 2 times 4.

8 + 3(-5) = 8 + (-15)

Step 3. Multiply 3 times $-5$ . A positive times a negative is negative. (See multiplying and dividing integers.)

8 + (-15) = -7

Step 4. Add. (See adding and subtracting integers.)

Answer: $-7$

Example 4: Evaluate with an exponent and a fraction

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

\frac{x^2 - 1}{y} + xy = \frac{(-3)^2 - 1}{2} + (-3)(2)

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

\frac{(-3)^2 - 1}{2} + (-3)(2) = \frac{9 - 1}{2} + (-3)(2)

Step 2. The fraction bar groups the top, so work on the top first. Square $-3$ , following the order of operations.

\frac{9 - 1}{2} + (-3)(2) = \frac{8}{2} + (-3)(2)

Step 3. Subtract on the top.

\frac{8}{2} + (-3)(2) = 4 + (-3)(2)

Step 4. Divide.

4 + (-3)(2) = 4 + (-6)

Step 5. Multiply.

4 + (-6) = -2

Step 6. Add.

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.

11t

Step 1. Write the cost of the tickets (dollars).

6p

Step 2. Write the cost of the popcorn (dollars).

\text{cost} = 11t + 6p

Step 3. The total is the sum.

11(4) + 6(2)

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

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

Step 5. Multiply 11 times 4.

44 + 6(2) = 44 + 12

Step 6. Multiply 6 times 2.

44 + 12 = 56

Step 7. Add.

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

Common Mistakes

Mistake	What it looks like	How to fix it
Flipping “less than”	“5 less than $x$ ” written as $5 - x$	Start with $x$ and take away 5: $x - 5$ .
Leaving out parentheses	“Twice the sum of $x$ and 3″ written as $2x + 3$	Double the whole sum: $2(x + 3)$ .
Squaring a negative without parentheses	If $x = -4$ , then $x^2 = -16$	$(-4)^2 = 16$
Adding before multiplying	$3 + 2x$ at $x = 5$ becomes $5 \cdot 5 = 25$	Multiply 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.

$n$	Cost
1	13
2	21
3	29
4	37

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.

How many terms are in $6x^2 - x + 4y - 10$ ? List them.
Write an expression for “the product of 9 and a number.”
Write an expression for “12 less than a number.”
Write an expression for “the sum of a number and 6, divided by 5.”
Write an expression for “8 more than twice a number.”
Evaluate $5x - 2y$ when $x = 3$ and $y = 4$ .
Evaluate $a^2 - 3a + 2$ when $a = -2$ .
Evaluate $\frac{m + n}{m - n}$ when $m = 7$ and $n = 3$ .
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.
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

6x^2 + (-x) + 4y + (-10)

Rewrite with plus signs.

Count the parts between the plus signs: 4 terms

6x^2,\ -x,\ 4y,\ -10

List them.

Let $n$ = the number

9n

“The product of 9 and” means 9 times.

Let $n$ = the number

n - 12

“12 less than” means take 12 away from the number.

Let $n$ = the number

n + 6

Write the sum.

\frac{n + 6}{5}

Divide the whole sum by 5.

Let $n$ = the number

2n

“Twice a number”.

2n + 8

“8 more than” means add 8.

5(3) - 2(4)

Replace x with 3 and y with 4.

15 - 2(4)

Multiply 5 times 3.

15 - 8

Multiply 2 times 4.

7

Subtract.

(-2)^2 - 3(-2) + 2

Replace a with −2.

4 - 3(-2) + 2

Do the exponent.

4 - (-6) + 2

Multiply.

4 + 6 + 2

Subtracting a negative is adding a positive.

10 + 2

Add 4 and 6.

12

Add.

\frac{7 + 3}{7 - 3}

Replace m with 7 and n with 3.

\frac{10}{7 - 3}

Add on the top.

\frac{10}{4}

Subtract on the bottom.

\frac{5}{2}

Simplify by dividing the top and bottom by 2 (see fractions)

2m + 3.50

Write the expression: $2 per mile plus $3.50.

2(12) + 3.50

Replace m with 12.

24 + 3.50

Multiply.

27.50

Add.

The ride costs $27.50.

10.

4n

Find the change: the value goes up by 4 each time, so start with.

4(1) = 4

Test n = 1.

4n + 3

The first value is 7, which is 3 more than 4, so add 3.

Check n = 4: $4(4) + 3 = 16 + 3 = 19$

Want more practice? The Algebra 1 Unit 1 Practice Pack has 26 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

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

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