Identify the property that justifies each step asked about in the answer area below.

Line 1: , left parenthesis, x, plus, 6, right parenthesis, left parenthesis, 7, plus, 4, x, right parenthesis

Line 1: =

(*x*+6)(7+4*x*)

Line 2: , left parenthesis, x, plus, 6, right parenthesis, left parenthesis, 4, x, plus, 7, right parenthesis

Line 2: =

(*x*+6)(4*x*+7)

Line 3: , left parenthesis, 4, x, plus, 7, right parenthesis, left parenthesis, x, plus, 6, right parenthesis

Line 3: =

(4*x*+7)(*x*+6)

Let's analyze each step and identify the mathematical property used in the transformation:

**Line 1 to Line 2:**

Expression changes from:  
\[
(x + 6)(7 + 4x)
\]

to:  
\[
(x + 6)(4x + 7)
\]

**Property used: Commutative Property of Addition**  
**Explanation:** The commutative property of addition states that \(a + b = b + a\). In this step, \(7 + 4x\) was rewritten as \(4x + 7\). The order of the terms inside the second set of parentheses was changed.

---

**Line 2 to Line 3:**

Expression changes from:  
\[
(x + 6)(4x + 7)
\]

to:  
\[
(4x + 7)(x + 6)
\]

**Property used: Commutative Property of Multiplication**  
**Explanation:** The commutative property of multiplication states that \(a \cdot b = b \cdot a\). In this step, the two binomials \((x + 6)\) and \((4x + 7)\) were swapped in position. The order of multiplication was changed.

### Summary of Properties Used:
1. **Line 1 to Line 2:** Commutative Property of Addition
2. **Line 2 to Line 3:** Commutative Property of Multiplication

Would you like more details on any of the properties?

### Related Questions:
1. What is the commutative property, and how does it differ for addition and multiplication?
2. Can the associative property be applied in this scenario? Why or why not?
3. What other properties of real numbers are useful for simplifying algebraic expressions?
4. How would these properties apply if subtraction or division were involved instead of addition or multiplication?
5. Can you provide an example where the distributive property is used to expand a similar expression?

**Tip:** Remember that the commutative property applies only to addition and multiplication, not subtraction or division.

Explore how the commutative properties of addition and multiplication are applied in algebraic expressions. This detailed explanation covers transformations from one expression to another using these fundamental properties. Suitable for high school students.

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