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:

$3x$ and $5x$ are like terms (both have $x$ )
$2x^2$ and $-4x^2$ are like terms (both have $x^2$ )
$7$ and $-3$ are like terms (both are constants)

Not like terms:

$3x$ and $3x^2$ (different powers)
$2x$ and $2y$ (different variables)

Simplifying process:

3x + 5x = 8x

2x^2 + 4x - x^2 + 3x = x^2 + 7x

Try it now

Which terms are like terms?

Worked Examples

Simplify: $7x + 4x$

Identify like terms

Both terms have the variable $x$ → $7x$ and $4x$ are like terms

Add the coefficients

$7 + 4 = 11$ → New coefficient is $11$

Write the simplified expression

$7x + 4x = 11x$

Answer: $7x + 4x = 11x$

Common Mistakes

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

Why it's wrong: Different variables cannot be combined. $x$ and $y$ represent different unknown values.

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

Adding different powers: $2x + 3x^2 = 5x^2$ or $5x^3$

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

Correct: $2x + 3x^2$ is already simplified. Write in standard form: $3x^2 + 2x$

Forgetting negative signs: $5x - 3x = 8x$

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

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

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

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

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

Interactive Visual

Balance Scale

x + 3=7

Apply to both sides:

Solution: x = 4

Interactive Sandbox

Expression Calculator

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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 $5x = 20$ is much easier to solve than $2x + 3x = 20$
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 $x$ dollars each, buying 3 shirts and then 2 more shirts costs $3x + 2x = 5x$ dollars total.

1Try It Yourself

You buy 4 notebooks at $n$ euros each, 3 pens at $p$ euros 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 $2x$ and width $x$ . Perimeter = $2x + x + 2x + x = 6x$

2Try It Yourself

A triangle has sides of length $3a$ , $2a + 5$ , and $4a - 3$ .

Find 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

x

and

y

represent different unknown values.

3x

means '3 times some number x' and

5y

means '5 times some different number y'. Without knowing what

x

and

y

are, we can't combine them into one term.

Glossary

Like terms: Terms that have the same variable(s) raised to the same power(s). Example: $3x$ and $7x$ are like terms.
Coefficient: The number multiplied by a variable. In $5x$ , the coefficient is $5$ .
Constant: A term without a variable. Example: $7$ or $-3$ .
Variable: A letter that represents an unknown number. Example: $x$ , $y$ , $a$ .
Standard form: Writing terms in order from highest power to lowest, with the constant last.