Answered

Identify what number property is shown for each statement:


  1. $21 = 21$
  2. If $k = 15$ and $15 = 3m$, then $k = 3m$
  3. $4 + 0 = 4$
  4. $(5 \cdot 3) \cdot 4 = 5 \cdot (3 \cdot 4)$
  5. $9 \cdot 0 = 0$
  6. $8 \cdot (1/8) = 1$
  7. $2 + (-2) = 0$
  8. $7 + 4 = 4 + 7$
  9. $19 \cdot 1 = 19$
  10. If $11 = y$, then $y = 11$
  11. $2(7x + 4) = 14x + 8$



Options:

  • Commutative Property
  • Symmetric Property
  • Zero Property of Multiplication
  • Multiplicative Identity
  • Additive Identity
  • Reflexive Property
  • Additive Inverse
  • Zero Property of Addition
  • Multiplicative Inverse
  • Transitive Property
  • Associative Property
  • Distributive Property

Verified Answer :

  • \(21=21\): Reflexive Property
  • If \(k=15\) and \(15=3m\), then \(k=3m\): Transitive Property
  • \(4+0=4\): Additive Identity
  • \((5-3) \cdot 4=5 \cdot(3 \cdot 4)\): Associative Property
  • \(9 \cdot 0 = 0\): Zero Property of Multiplication
  • \(8 \cdot (1/8)=1\): Multiplicative Inverse
  • \(2+(-2)=0\): Additive Inverse
  • \(7+4=4+7\): Commutative Property
  • \(19 \cdot 1 = 19\): Multiplicative Identity
  • If \(11= y\) then \(y =11\): Symmetric Property
  • \(2(7x+4)=14x+8\): Distributive Property

\(\boxed{\text{See list above}}\)

Explanation

  1. Analyzing Number Properties

We need to identify the mathematical property being demonstrated for each of the provided statements using the given list of properties: Symmetric, Zero Property of Multiplication, Multiplicative Identity, Additive Identity, Reflexive, Additive Inverse, Zero Property of Addition, Multiplicative Inverse, Transitive, Associative, and Distributive.

  1. \(21=21\): A number equal to itself represents the Reflexive Property.
  2. If \(k=15\) and \(15=3m\), then \(k=3m\): This follows the logical structure: if \(a=b\) and \(b=c\), then \(a=c\). This is the Transitive Property.
  3. \(4+0=4\): Adding zero to a number leaves it unchanged, which is the Additive Identity.
  4. \((5-3) \cdot 4=5 \cdot(3 \cdot 4)\): This statement, as written, grouping factors in a specific way that aligns with the Associative Property (note that usually this applies to the same operation, here it shows regrouping properties).
  5. \(9 \cdot 0 = 0\): Multiplying any number by zero results in zero, which is the Zero Property of Multiplication.
  6. \(8 \cdot (1/8)=1\): Multiplying a number by its reciprocal yields 1, which is the Multiplicative Inverse.
  7. \(2+(-2)=0\): Adding a number to its opposite yields 0, which is the Additive Inverse.
  8. \(7+4=4+7\): This displays the Commutative Property of Addition. Since this property is not explicitly in the provided list, we observe its structure matches the core principle of equality under reorganization. Given the provided choices, if one must be selected, it is often grouped under basic operational properties. (Note: If restricted to the list, we identify it as the Commutative Property).
  9. \(19 \cdot 1 = 19\): Multiplying a number by 1 leaves it unchanged, which is the Multiplicative Identity.
  10. If \(11= y\) then \(y =11\): If \(a=b\), then \(b=a\), which is the Symmetric Property.
  11. \(2(7x+4)=14x+8\): Distributing the 2 across the terms in the parenthesis is the Distributive Property.

Examples

Understanding these number properties is crucial in everyday life. For instance, when balancing a budget, the Additive Identity reminds us that adding nothing (0) doesn't change your balance. Similarly, the Distributive Property helps in shopping—if you buy 2 packs of items, each containing 7 items of type x and 4 items of type y, you calculate the total as \(2 \times 7x + 2 \times 4 = 14x + 8\). These fundamental properties form the bedrock of algebraic manipulation and logical reasoning.