Foundations and Expressions: Algebra 1 Chapter 1

Why this chapter matters

Every later equation, function, and model depends on reading symbols correctly and combining only terms that truly match.

What you will learn

Evaluate expressions using substitution and the order of operations.
Apply distributive, commutative, and associative properties.
Combine like terms and translate verbal statements into algebra.

Understand the core ideas

An algebraic expression is a compact way to describe repeated calculation. Each term carries a numeric part and, sometimes, a variable part. Terms can be added or subtracted only when their variable parts match exactly. For example, 4x and 9x combine because both measure the same unit of x, but 4x and $9x^2$ do not. Properties such as commutative and associative let you reorder and regroup without changing value, which is useful when you want like terms to sit next to each other before simplifying.

The distributive property links multiplication and addition, and it is one of the most important moves in early algebra. If a package has 3 groups of (2x - 5), distributing gives 6x - 15, which is easier to combine with other terms. Substitution checks whether two forms are equivalent by plugging in a number for the variable and comparing results. When translating words into expressions, mark each quantity first, then write operations in the order the sentence implies. This habit prevents common sign mistakes and missing terms.

Key terms

term: A single part of an expression separated by plus or minus signs.
coefficient: The numeric factor multiplying a variable, such as 7 in 7x.
like terms: Terms with identical variable parts and exponents that can be combined.
distributive property: Rule that multiplies a factor across a sum or difference, a(b + c) = ab + ac.

Simplify a multi-part expression

Simplify 2(3x - 4) + 5x - (x - 6).

Distribute in the first group: 2(3x - 4) becomes 6x - 8.
Distribute the subtraction across the second group: -(x - 6) becomes -x + 6.
Rewrite the full expression as 6x - 8 + 5x - x + 6.
Combine like terms: (6x + 5x - x) = 10x and (-8 + 6) = -2, so the result is 10x - 2.

Result: The simplified expression is 10x - 2.

A common misconception

Claim: You can combine any terms that use x, so $3x + 4x^2 = 7x^3$ .

Correction: Only like terms combine. Since x and $x^2$ have different exponents $, 3x + 4x^2$ stays as two terms.

Lessons in this chapter

Variables and expressionsIdentify coefficients, constants, terms, and operations in an expression.
Properties of real numbersRewrite expressions while preserving equality.
Combining like termsSimplify expressions by grouping terms with the same variable part.
Writing expressions from contextRepresent quantities and relationships with algebraic notation.

Study task

Write three different expressions for a phone plan with a fixed monthly fee and a per-gigabyte charge, then explain what each term means.

Chapter checkpoint

Simplify 3(2x - 5) + 4x.

Distribute first: 6x - 15 + 4x. Combine like terms to get 10x - 15.

Foundations and Expressions