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Properties of Operations

A free Pre-Algebra lesson from the “Algebra Readiness” unit, with a worked example and practice problems including step-by-step solutions.

The commutative property says order does not matter for addition or multiplication (a + b = b + a, a * b = b * a). The associative property says grouping does not matter ((a + b) + c = a + (b + c)). The identity property: a + 0 = a and a * 1 = a. The distributive property: a(b + c) = ab + ac.

What you'll learn

Why it matters: Mental-math shortcuts (regrouping 4 x 25 x 7 as (4 x 25) x 7 = 100 x 7) and every step of algebraic simplification rely on these properties — they are the rules behind why algebra moves are legal.

Worked example

Problem. Use the distributive property to compute 6(20 + 3).

  1. Distributive: a(b + c) = ab + ac.
  2. 6(20) + 6(3) = 120 + 18.
  3. 120 + 18 = 138.

Answer: 138

Practice problems

1. Which property is shown by 3 + 7 = 7 + 3?

Choices: Commutative · Associative · Identity · Distributive

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which property is shown by 3 + 7 = 7 + 3?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The order of the two addends switched but the values stayed the same.
  4. That is the commutative property of addition.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Commutative

2. Which property is shown by 2(5x) = 10x?

Choices: Commutative · Associative · Identity · Distributive

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which property is shown by 2(5x) = 10x?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The grouping (2 * 5) * x became 10 * x.
  4. That regrouping is the associative property of multiplication.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Associative

3. Use the distributive property to expand 5(x + 4).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Use the distributive property to expand 5(x + 4).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply 5 by each term inside.
  4. 5 * x + 5 * 4 = 5x + 20.
  5. Check the result by substituting or estimating: the response should match 5x + 20 and make sense in the original problem.

Answer: 5x + 20

4. Expand 3(2x - 7).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Expand 3(2x - 7).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Distribute the 3: 3 * 2x - 3 * 7.
  4. = 6x - 21.
  5. Check the result by substituting or estimating: the response should match 6x - 21 and make sense in the original problem.

Answer: 6x - 21

5. Use the commutative property: 4 + x = ___ + 4. Enter what fills the blank.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use the commutative property: 4 + x = ___ + 4. Enter what fills the blank.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Commutative lets you swap order.
  4. 4 + x = x + 4.
  5. Check the result by substituting or estimating: the response should match x and make sense in the original problem.

Answer: x

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