Which property does each equation demonstrate?
[tex]x^2+2 x=2 x+x^2[/tex]
[tex]\left(3 z^4+2 z^3\right)-\left(2 z^4+z^3\right)=z^4+z^3[/tex]
[tex]\left(2 x^2+7 x\right)+\left(2 y^2+6 y\right)=\left(2 y^2+6 y\right)+\left(2 x^2+7 x\right)[/tex]
Answer (2)
The first and third equations demonstrate the commutative property of addition, while the second equation shows simplification by combining like terms. Understanding these properties is crucial for working with algebraic expressions. The properties allow for flexibility in how we manage and compute expressions.
;
Equation 1 demonstrates the commutative property of addition: x 2 + 2 x = 2 x + x 2 .
Equation 2 demonstrates simplification by combining like terms: ( 3 z 4 + 2 z 3 ) − ( 2 z 4 + z 3 ) = z 4 + z 3 .
Equation 3 demonstrates the commutative property of addition: ( 2 x 2 + 7 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) .
Therefore, the properties demonstrated are commutative property of addition and simplification by combining like terms.
Explanation
Problem Analysis We are given three equations and asked to identify the property that each equation demonstrates. Let's analyze each equation separately.
Analyzing Equation 1 Equation 1: x 2 + 2 x = 2 x + x 2 . This equation shows that the order in which we add x 2 and 2 x does not affect the result. This is an example of the commutative property of addition.
Analyzing Equation 2 Equation 2: ( 3 z 4 + 2 z 3 ) − ( 2 z 4 + z 3 ) = z 4 + z 3 . To determine the property demonstrated, we simplify the left side of the equation:
( 3 z 4 + 2 z 3 ) − ( 2 z 4 + z 3 ) = 3 z 4 + 2 z 3 − 2 z 4 − z 3 = ( 3 z 4 − 2 z 4 ) + ( 2 z 3 − z 3 ) = z 4 + z 3 .
This equation demonstrates simplification by combining like terms.
Analyzing Equation 3 Equation 3: ( 2 x 2 + 7 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) . This equation shows that the order in which we add the expressions ( 2 x 2 + 7 x ) and ( 2 y 2 + 6 y ) does not affect the result. This is another example of the commutative property of addition.
Conclusion In summary:
Equation 1 demonstrates the commutative property of addition.
Equation 2 demonstrates simplification by combining like terms.
Equation 3 demonstrates the commutative property of addition.
Examples
The commutative property is useful in everyday situations. For example, if you are buying items at a store, the order in which the cashier adds up the prices doesn't change the total amount you owe. Similarly, combining like terms is essential in simplifying complex expressions in various fields, such as physics and engineering, where equations often need to be simplified to make calculations easier.
