Which property does each equation demonstrate?

*x*2 + 2*x* = 2*x* + *x*2
associative propertyclosure propertycommutative property
 
(3*z*4 + 2*z*3) – (2*z*4 + *z*3) = *z*4 + *z*3
associative propertyclosure propertycommutative property
 
(2*x*2 + 7*x*) + (2*y*2 + 6*y*) = (2*y*2 + 6*y*) + (2*x*2 + 7*x*)

Let's analyze each equation and identify the property it demonstrates:

1. **Equation:** \( x^2 + 2x = 2x + x^2 \)
   
   This equation shows that changing the order of the terms does not affect the sum. This is the **commutative property** of addition.

2. **Equation:** \( (3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3 \)
   
   This equation simplifies the expression on both sides and demonstrates that the sum and difference of polynomials is also a polynomial. This is the **closure property**.

3. **Equation:** \( (2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x) \)
   
   This equation shows that the sum of two polynomials is independent of the order in which they are added. This is the **commutative property** of addition.

In summary:
1. \( x^2 + 2x = 2x + x^2 \) demonstrates the **commutative property**.
2. \( (3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3 \) demonstrates the **closure property**.
3. \( (2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x) \) demonstrates the **commutative property**.

Would you like more details or have any questions?

Here are 5 more questions for you to explore:
1. What is the associative property of addition?
2. How does the distributive property work with multiplication over addition?
3. Can you provide an example of the associative property with three numbers?
4. What is the closure property in the context of multiplication?
5. How do you identify the commutative property in subtraction?

**Tip:** When trying to identify properties of equations, focus on the operation (addition, multiplication, etc.) and the arrangement of terms to see if their order or grouping changes the outcome.

Explore properties of equations such as commutative and closure properties with detailed examples. Learn how these properties apply to polynomials and algebraic expressions. Suitable for Grades 7-9.

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