Simplifying Expressions

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, and multiplication) with no equals sign. For example, 3x + 5y - 2x + 7 is an expression. Our goal in this lesson is to write expressions in their simplest possible form by combining parts that behave the same way.

Terms, Coefficients, and Like Terms

An expression is built out of terms — pieces separated by a plus or minus sign. In 3x + 5y - 2x + 7, the terms are 3x, 5y, -2x, and 7.

• The number multiplying a variable is called the coefficient. In 3x, the coefficient is 3.
• A term with no variable, like 7, is called a constant.
• Like terms have the exact same variable part (same letters raised to the same powers). 3x and -2x are like terms; 5y is not like either of them, since it has a different variable.

Combining Like Terms

To simplify an expression, add or subtract the coefficients of like terms and leave the variable part unchanged. Think of x as a placeholder for "some object" — 3 apples minus 2 apples is 1 apple, and in the same way 3x - 2x = x. You can only combine terms that share the same variable part; you can never combine an x term with a y term or with a plain number.

Example

Simplify 3x + 5y - 2x + 7 - y.

1. Group like terms together: (3x - 2x) + (5y - y) + 7.
2. Combine the x terms: 3x - 2x = x.
3. Combine the y terms: 5y - y = 4y.
4. The constant 7 has nothing to combine with, so it stays as is.

Putting it together, the simplified expression is x + 4y + 7

A Word of Caution

Watch the sign attached to each term — it travels with the term. In 5y - y, remember that a lone y means 1y, so the result is 4y, not 5. Also, exponents matter for "sameness": x and x^2 are not like terms, even though they share the same letter.

Key Takeaways

• Terms are the pieces of an expression separated by + or -.
• Like terms have identical variable parts; only their coefficients get combined.
• A sign in front of a term belongs to that term when you rearrange the expression.
• Simplifying never changes what the expression equals — it just writes it more compactly.