Simplifying Algebraic Expressions

Learn how to simplify algebraic expressions by combining like terms and using properties of operations.

Intermediate25 minLesson
Printable worksheet

Definition

Simplifying an algebraic expression means rewriting it in its simplest form while keeping the same value.
To simplify, we combine like terms:
  • Like terms have the same variable raised to the same power
  • We add or subtract their coefficients (the numbers in front)
Examples of like terms:
  • 3x3x and 5x5x are like terms (both have xx)
  • 2x22x^2 and −4x2-4x^2 are like terms (both have x2x^2)
  • 77 and −3-3 are like terms (both are constants)
Not like terms:
  • 3x3x and 3x23x^2 (different powers)
  • 2x2x and 2y2y (different variables)
Simplifying process:
3x+5x=8x3x + 5x = 8x
2x2+4x−x2+3x=x2+7x2x^2 + 4x - x^2 + 3x = x^2 + 7x

Try it now

Which terms are like terms?

Worked Examples

Simplify: 7x+4x7x + 4x
1

Identify like terms

Both terms have the variable xx → 7x7x and 4x4x are like terms
2

Add the coefficients

7+4=117 + 4 = 11 → New coefficient is 1111
3

Write the simplified expression

7x+4x=11x7x + 4x = 11x

Common Mistakes

Combining unlike terms: 3x+4y=7xy3x + 4y = 7xy

Why it's wrong: Different variables cannot be combined. xx and yy represent different unknown values.

Correct: 3x+4y3x + 4y is already simplified - these terms cannot be combined.

Adding different powers: 2x+3x2=5x22x + 3x^2 = 5x^2 or 5x35x^3

Why it's wrong: xx and x2x^2 are not like terms because they have different exponents.

Correct: 2x+3x22x + 3x^2 is already simplified. Write in standard form: 3x2+2x3x^2 + 2x

Forgetting negative signs: 5x−3x=8x5x - 3x = 8x

Why it's wrong: Subtraction means adding a negative: 5x−3x=5x+(−3x)5x - 3x = 5x + (-3x)

Correct: 5x−3x=2x5x - 3x = 2x (we subtract 3 from 5)

Multiplying instead of adding coefficients: 4x+2x=8x4x + 2x = 8x

Why it's wrong: When combining like terms, we add (or subtract) the coefficients, not multiply them.

Correct: 4x+2x=6x4x + 2x = 6x (add 4 and 2 to get 6)

Interactive Visual

Algebra tiles: combining like terms

Each tile is a term: big squares are x², long strips are x, small squares are 1. Red tiles are negative.

4x4x
77
−2x-2x
33

4x+7−2x+34x + 7 - 2x + 3

Every term becomes tiles: the coefficient tells how many.

=x2= x^2=x= x=1= 1negative

Function graph

-6-4-20246-50510152025xy

Tap or hover over the graph to read off values.

4x + 7 - 2x + 32x + 10

Table of values

x0123
4x + 7 - 2x + 310121416
2x + 1010121416

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems
Problem 1 of 16
Easy
Which terms are like terms?

Why It Matters

Simplifying expressions is a fundamental skill that makes algebra much easier:
  • Solve equations faster: A simplified equation like 5x=205x = 20 is much easier to solve than 2x+3x=202x + 3x = 20
  • See patterns clearly: Simplified expressions reveal mathematical relationships
  • Reduce calculation errors: Fewer terms mean fewer chances for mistakes
  • Real-world applications: From calculating costs to engineering formulas, simplified expressions save time
Think of simplifying like cleaning your room - everything becomes easier to find and work with!

Real World Applications

Shopping Calculations

When buying multiple items, simplifying helps find the total cost quickly.

Example:

If shirts cost xx dollars each, buying 3 shirts and then 2 more shirts costs 3x+2x=5x3x + 2x = 5x dollars total.

1Try It Yourself

You buy 4 notebooks at nn dollars each, 3 pens at pp dollars each, then return 1 notebook.

Write and simplify the expression for your total cost.

Step 1: Write the mathematical expression

Start with items bought, then subtract the return:

Perimeter of Shapes

Finding perimeters often requires simplifying expressions with variables.

Example:

A rectangle has length 2x2x and width xx. Perimeter = 2x+x+2x+x=6x2x + x + 2x + x = 6x

2Try It Yourself

A triangle has sides of length 3a3a, 2a+52a + 5, and 4a−34a - 3.

Write and simplify the expression for the perimeter.

Step 1: Write the mathematical expression

Add all three sides:

Key Takeaways

  • 1Like terms have the same variable(s) with the same exponent(s)
  • 2To simplify, add or subtract the coefficients of like terms
  • 3Constants (numbers without variables) are like terms with each other
  • 4Terms with different variables or different exponents cannot be combined
  • 5Write final answers in standard form (highest power first)

Frequently Asked Questions

Because xx and yy represent different unknown values. 3x3x means '3 times some number x' and 5y5y means '5 times some different number y'. Without knowing what xx and yy are, we can't combine them into one term.
Because xx and yy represent different unknown values. 3x3x means '3 times some number x' and 5y5y means '5 times some different number y'. Without knowing what xx and yy are, we can't combine them into one term.
Yes! When a variable has no visible coefficient, the coefficient is 1. So x=1xx = 1x, and when simplifying, we count xx as having coefficient 1.
x2x^2 means 'xx times xx' while xx is just 'xx'. They represent different quantities. For example, if x=3x = 3, then x=3x = 3 but x2=9x^2 = 9. That's why 2x+3x22x + 3x^2 cannot be simplified further.
If all terms cancel, the answer is 00. For example, 5x−5x=05x - 5x = 0. This is a valid simplified result!

Glossary

Like terms
Terms that have the same variable(s) raised to the same power(s). Example: 3x3x and 7x7x are like terms.
Coefficient
The number multiplied by a variable. In 5x5x, the coefficient is 55.
Constant
A term without a variable. Example: 77 or −3-3.
Variable
A letter that represents an unknown number. Example: xx, yy, aa.
Standard form
Writing terms in order from highest power to lowest, with the constant last.
Today's challengeOne problem, three tries

More in This Topic

Variables and Algebraic Expressions

Intermediate25 min

Evaluating Expressions

Intermediate25 min

Writing Algebraic Expressions

Intermediate25 min

Combining Like Terms

Intermediate25 min