Which property does each equation demonstrate?
Associative property
[tex]z^4+z^3[/tex]
Closure property
Commutative property
[tex](2 x^2+1 x)+(2 y^2+6 y)=(2 y^2+6 y)+(2 x^2+7 x)[/tex]
Answer (2)
The first equation z 4 + z 3 does not demonstrate any specific property. The second equation does not accurately show the commutative property because the expressions on each side are not equal. Both equations highlight the importance of understanding how mathematical properties function in expressions and equations.
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The commutative property states that a + b = b + a .
The given equation ( 2 x 2 + 1 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) does not demonstrate the commutative property because 1 x = 7 x .
If the equation was ( 2 x 2 + x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + x ) , then it would demonstrate the commutative property.
The original equation does not directly demonstrate any of the listed properties due to the inequality of expressions on both sides.
Explanation
Understanding the Problem We are asked to identify the property demonstrated by the equation ( 2 x 2 + 1 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) . The possible properties are associative, closure, and commutative.
Recalling Properties The commutative property states that a + b = b + a . The associative property states that ( a + b ) + c = a + ( b + c ) . The closure property states that if a and b are in a set, then a + b is also in the set.
Analyzing the Equation The given equation is ( 2 x 2 + 1 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) . Notice that the order of the terms being added is changed. However, the expressions on both sides are not equal because 1 x = 7 x . Therefore, the equation does not demonstrate the commutative property.
Considering a Modified Equation Let's consider a modified equation: ( 2 x 2 + x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + x ) . This equation demonstrates the commutative property because the order of the terms being added is changed, and the expressions on both sides are equal.
Conclusion However, the original equation ( 2 x 2 + 1 x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + 7 x ) does not demonstrate any of the listed properties directly because the expressions on both sides are not equal. There seems to be a typo in the original equation. If the equation was ( 2 x 2 + x ) + ( 2 y 2 + 6 y ) = ( 2 y 2 + 6 y ) + ( 2 x 2 + x ) , then it would demonstrate the commutative property.
Examples
The commutative property is fundamental in many areas of mathematics and everyday life. For example, when calculating the total cost of items, the order in which you add the prices doesn't change the final amount. If you buy a shirt for $25 and pants for $40, the total cost is $25 + $40 = $65, which is the same as $40 + $25 = $65. This principle applies to various situations, from managing finances to understanding basic arithmetic.
