To determine the property each equation demonstrates, let's examine them one by one:
1. Equation 1: [tex]\( x^2 + 2x = 2x + x^2 \)[/tex]
This equation shows that the order in which two terms are added does not affect the sum. This concept is known as the Commutative Property of Addition. Specifically, it states that for any numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex], [tex]\(a + b = b + a\)[/tex].
Hence, Equation 1 demonstrates the Commutative Property of Addition.
2. Equation 2: [tex]\( (3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3 \)[/tex]
In this equation, we first combine like terms within each polynomial and then perform the subtraction operation. This process involves distributing the minus sign and combining like terms, which is an application of the Distributive Property. The Distributive Property states that [tex]\( a(b - c) = ab - ac \)[/tex] and allows us to rearrange and simplify expressions by distributing a common factor.
Hence, Equation 2 demonstrates the Distributive Property.
3. Equation 3: [tex]\( (2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x) \)[/tex]
Similar to the first equation, this equation rearranges the order of two groups of terms being added. This, once again, represents the Commutative Property of Addition because it shows that the sum remains unchanged regardless of the order in which the terms are added.
Hence, Equation 3 demonstrates the Commutative Property of Addition.
To summarize:
- Equation 1: Commutative Property of Addition
- Equation 2: Distributive Property
- Equation 3: Commutative Property of Addition
