How to Justify Each Step Using Commutativity and Associativity?

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In summary, commutativity and associativity are important properties in an equation, as they allow us to rearrange and group terms without altering the result. This makes solving complex equations easier and more efficient, and can also help in simplifying expressions and identifying patterns. An example of using commutativity and associativity in an equation is (3-2)+(7-4) = (3+7)+(-2-4).
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happyprimate
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Exercise 3 Chapter 1 Basic Mathematics Serge Lang

Verifying my answer.

My answer:

(a-b)+(c-d) = (a+c)+(-b-d)

Let p = (a-b)+(c-d) We need to show that p = (a+c)+(-b-d)

(a-b)+(c-d)

a+(-b+(c-d)) Associativity

a+((-b+c)-d) Associativity

a+((c-b)-d) Commutativity

((a+c)-b)-d) Associativity

(a+c)+(-b-d) Associativity
 
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Looks good to me.
 

FAQ: How to Justify Each Step Using Commutativity and Associativity?

How do commutativity and associativity apply in the equation (a-b)+(c-d) = (a+c)+(-b-d)?

Commutativity allows us to rearrange the order of the terms in an addition or multiplication equation without changing the result. In this case, we can swap the positions of (a-b) and (c-d) to get (c-d)+(a-b). Associativity, on the other hand, allows us to group terms in different ways without changing the result. This means we can group (a-b) and (c-d) as (a+c)-(b+d) or (a+c)+(-b-d). Both of these properties are used in the given equation to show that the two sides are equivalent.

Why is it important to justify each step in the equation using commutativity and associativity?

Justifying each step using commutativity and associativity ensures that we are following the correct mathematical rules and that our solution is valid. It also helps to build a logical and clear argument for our solution, making it easier for others to understand and follow our reasoning.

Can we use commutativity and associativity in any type of equation?

Commutativity and associativity can only be applied to addition and multiplication equations. They do not apply to other mathematical operations such as subtraction, division, or exponentiation.

How do commutativity and associativity differ from the distributive property?

The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. This is different from commutativity and associativity, which only deal with rearranging or grouping terms in an equation.

Are there any other properties or rules that can be used to justify steps in an equation?

Yes, there are other properties and rules in mathematics that can be used to justify steps in an equation. Some examples include the identity property, which states that any number multiplied by 1 is equal to that number, and the inverse property, which states that any number added to its inverse (opposite) is equal to 0. These, along with other properties, can be used to simplify equations and prove their validity.

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