Properties of Real Numbers worksheet for algebra practice.

The image you provided is a worksheet titled "Properties of Real Numbers." The task involves identifying the correct property that corresponds to each given equation or statement. Below, I will solve the problem by matching each equation with its corresponding property and explaining the solution.

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List of Properties of Real Numbers

Before we begin, let's briefly review the properties of real numbers that are relevant to this worksheet:

1. Commutative Property of Addition: $a + b = b + a$
2. Commutative Property of Multiplication: $a \cdot b = b \cdot a$
3. Associative Property of Addition: $(a + b) + c = a + (b + c)$
4. Associative Property of Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
5. Identity Property of Addition: $a + 0 = a$
6. Identity Property of Multiplication: $a \cdot 1 = a$
7. Inverse Property of Addition: $a + (-a) = 0$
8. Inverse Property of Multiplication: $a \cdot \frac{1}{a} = 1$ (for $a \neq 0$)
9. Distributive Property: $a(b + c) = ab + ac$
10. Reflexive Property: $a = a$
11. Symmetric Property: If $a = b$, then $b = a$
12. Transitive Property: If $a = b$ and $b = c$, then $a = c$

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Solutions to the Worksheet

#### Section 1: Identify the Correct Property

1. 6.4 + 1 = 6.4
- Property: Identity Property of Addition
- Explanation: Adding 0 to any number leaves it unchanged. Here, $6.4 + 0 = 6.4$.

2. 3x(2 + 5) = 3x(7)
- Property: Substitution Property
- Explanation: Simplifying $2 + 5$ to $7$ is a basic arithmetic operation, which is a form of substitution.

3. 5 + (4 + 2) = 5 + (2 + 4)
- Property: Commutative Property of Addition
- Explanation: The order of addition can be changed without affecting the result.

4. 7 - 2 = 5
- Property: Subtraction Property
- Explanation: This is a basic subtraction operation.

5. 3m + 3n = 3n + 3m
- Property: Commutative Property of Addition
- Explanation: The order of terms being added can be swapped.

6. (2x + 10) + 3m = 2x + (10 + 3m)
- Property: Associative Property of Addition
- Explanation: The grouping of terms being added does not affect the result.

7. 8x + 2x = 10x
- Property: Distributive Property (or Combining Like Terms)
- Explanation: Factoring out the common factor $x$ results in $10x$.

8. $\frac{x}{1} = x$
- Property: Identity Property of Multiplication
- Explanation: Dividing any number by 1 leaves it unchanged, which is equivalent to multiplying by 1.

9. 6d + 2 = 2 + 6d
- Property: Commutative Property of Addition
- Explanation: The order of terms being added can be swapped.

10. If $2x = 8$, then $x = 4$
- Property: Division Property of Equality
- Explanation: Solving for $x$ by dividing both sides of the equation by 2.

11. (-1)(-6) = 3(-2)
- Property: Multiplicative Property of Equality
- Explanation: Both sides of the equation are equal after performing the multiplications.

12. $4 + 0 = 4$
- Property: Identity Property of Addition
- Explanation: Adding 0 to any number leaves it unchanged.

13. $a + (-a) = 0$
- Property: Inverse Property of Addition
- Explanation: Adding a number to its additive inverse results in 0.

14. $5c = 5c$
- Property: Reflexive Property
- Explanation: Any number is equal to itself.

15. $3m = 3m$
- Property: Reflexive Property
- Explanation: Any number is equal to itself.

16. If $a = b$ and $b = c$, then $a = c$
- Property: Transitive Property
- Explanation: If two quantities are equal to a third quantity, they are equal to each other.

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Final Answer

The solutions are matched as follows:
1. Identity Property of Addition
2. Substitution Property
3. Commutative Property of Addition
4. Subtraction Property
5. Commutative Property of Addition
6. Associative Property of Addition
7. Distributive Property
8. Identity Property of Multiplication
9. Commutative Property of Addition
10. Division Property of Equality
11. Multiplicative Property of Equality
12. Identity Property of Addition
13. Inverse Property of Addition
14. Reflexive Property
15. Reflexive Property
16. Transitive Property

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