Chapter 06 of 10

Algebraic Expressions

How can a variable describe any value in a pattern or situation?

A variable stands for an unknown or changing quantity. Algebraic expressions translate relationships into compact mathematical language. Evaluating substitutes a specific value; rewriting shows when two expressions are equivalent.

01

The big ideas

Understand these before you worry about speed.

Learn the parts of an expression

Terms are separated by addition or subtraction. A coefficient multiplies a variable, and a constant has no variable.

ExampleIn 5x−3y+8\displaystyle 5x-3y+8, the coefficients are 5\displaystyle 5 and −3\displaystyle -3, and the constant is 8\displaystyle 8.

Translate words in their stated order

Words such as sum, product, quotient, more than, and less than signal operations. Be especially careful: five less than x means x − 5.

ExampleSeven more than three times n\displaystyle n is 3n+7\displaystyle 3n+7.

Substitute with parentheses

Replace every occurrence of the variable with its value. Parentheses preserve negative signs and make the order of operations clear.

ExampleIf x=−2\displaystyle x=-2, then x2+3=(−2)2+3=7\displaystyle x^2+3=(-2)^2+3=7.

Use properties to create equivalent expressions

The distributive property removes grouping, and like terms combine when their variable parts match exactly.

Example3(x+4)+2x=3x+12+2x=5x+12\displaystyle 3(x+4)+2x=3x+12+2x=5x+12.
02

Worked example

Follow the reason for each step—not just the symbols.

Problem

Simplify 4(2x−3)+5−x\displaystyle 4(2x-3)+5-x.

  1. Distribute 4\displaystyle 4: 8x−12+5−x\displaystyle 8x-12+5-x.
  2. Combine variable terms: 8x−x=7x\displaystyle 8x-x=7x.
  3. Combine constants: −12+5=−7\displaystyle -12+5=-7.
Answer7x−7\displaystyle 7x-7
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Identify the coefficient of m\displaystyle m in −6m+11\displaystyle -6m+11.

Check answer−6\displaystyle -6
2

Write an expression for four less than twice p\displaystyle p.

Check answer2p−4\displaystyle 2p-4
3

Evaluate 3a2−1\displaystyle 3a^2-1 when a=−2\displaystyle a=-2.

Check answer11\displaystyle 11
4

Simplify 5(y+2)−2y\displaystyle 5(y+2)-2y.

Check answer3y+10\displaystyle 3y+10
04

Practice and tools

Use the resource that matches what you need next.

Worksheet

Evaluating Algebraic Expressions

Substitute positive and negative values into expressions with powers and multiple variables.

Open resource Worksheet

Order of Operations

Strengthen the evaluation skills expressions require.

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Patterns and Sequences

Connect changing quantities to numerical patterns.

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Multi-Step Word Problems

Translate situations into organized calculations.

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