MHB How to Justify Each Step Using Commutativity and Associativity?

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The discussion focuses on verifying the expression (a-b)+(c-d) = (a+c)+(-b-d) using the properties of associativity and commutativity. The user demonstrates each step of the transformation, applying associativity to regroup terms and commutativity to rearrange them. The final expression confirms that the original and transformed equations are equivalent. The verification process emphasizes the importance of these mathematical properties in simplifying expressions. Overall, the approach effectively illustrates how to justify each step in the equation.
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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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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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