Certainly! Let's analyze each equation and identify the mathematical property they demonstrate.
1. Equation: [tex]\( x^2 + 2x = 2x + x^2 \)[/tex]
- In this equation, we see that the terms [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] can swap places without changing the equality. This illustrates the property where the order of addition does not matter. This is known as the Commutative Property of Addition.
2. Equation: [tex]\( (3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3 \)[/tex]
- Here, we distribute the subtraction over the addition inside the parentheses, effectively subtracting similar terms from each other: [tex]\( 3z^4 - 2z^4 \)[/tex] and [tex]\( 2z^3 - z^3 \)[/tex]. The result is [tex]\( z^4 + z^3 \)[/tex]. This operation demonstrates the Distributive Property of Subtraction over Addition.
3. Equation: [tex]\( (2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x) \)[/tex]
- This equation shows that the groups of terms, [tex]\( 2x^2 + 7x \)[/tex] and [tex]\( 2y^2 + 6y \)[/tex], can be arranged in any order without affecting the sum. This again illustrates that the order of addition does not matter. This is another example of the Commutative Property of Addition.
In summary, the properties demonstrated by the given equations are:
1. Commutative property of addition
2. Distributive property of subtraction over addition
3. Commutative property of addition
