Examkin

Question

Difficulty: MediumOrder of Operations and Number Properties

A student simplifies the expression 7+3(x+4)−77 + 3(x + 4) - 7 in three steps as shown below:

Step 1: 7−7+3(x+4)7 - 7 + 3(x + 4)
Step 2: 0+3(x+4)0 + 3(x + 4)
Step 3: 3x+123x + 12

Which of the following lists the mathematical properties of real numbers that justify Step 1 and Step 3, respectively?

  1. Commutative property of addition; distributive propertyAnswer
  2. B
    Associative property of addition; distributive property
  3. C
    Commutative property of addition; associative property of multiplication
  4. D
    Associative property of addition; commutative property of multiplication
  5. E
    Distributive property; commutative property of addition

Answer

Commutative property of addition; distributive property
In Step 1, the student reorders the terms from 7+3(x+4)−77 + 3(x+4) - 7 to 7−7+3(x+4)7 - 7 + 3(x+4). Since changing the order of elements being added is governed by the commutative property of addition, this property justifies Step 1. In Step 3, the expression 3(x+4)3(x + 4) is expanded to 3x+123x + 12 by multiplying 33 by both xx and 44. This operation is justified by the distributive property. Combining these, the correct pair of properties is the commutative property of addition and the distributive property.

Step-by-Step Solution

1
Analyze the transition from the initial expression to Step 1: 7+3(x+4)−7→7−7+3(x+4)7 + 3(x + 4) - 7 \rightarrow 7 - 7 + 3(x + 4)
The terms are rearranged such that the constant term −7-7 is moved next to the initial term 77.
The commutative property of addition states that the order in which real numbers are added does not affect the sum (a+b=b+aa + b = b + a).
2
Analyze the transition from Step 2 to Step 3: 3(x+4)→3x+123(x + 4) \rightarrow 3x + 12
The coefficient 33 is multiplied by both terms inside the parentheses (xx and 44).
The distributive property states that multiplying a sum by a number is equivalent to multiplying each term in the sum individually (a(b+c)=ab+aca(b + c) = ab + ac).

Key Concept

Identifying commutative and distributive properties of real numbers.
Estimated Time:1m 0s
Rate this question

Topics

Order of Operations and Number Properties

More questions from this topic