To prove the equation $(x + y) + z = x + (y + z)$, we need to demonstrate that the grouping of the terms doesn't affect the result of the addition. This property is known as the associative property of addition.

Proof:

1. Let $x$, $y$, and $z$ be any real numbers.
2. We need to show that: $(x + y) + z = x + (y + z)$
3. Addition of real numbers follows the associative property, which states that for any real numbers $a$, $b$, and $c$, we have: $(a + b) + c = a + (b + c)$
4. By setting $a = x$, $b = y$, and $c = z$, we can directly apply this property: $(x + y) + z = x + (y + z)$ This equality holds because the way in which numbers are grouped during addition does not affect the outcome.

Thus, we have proven that $(x + y) + z = x + (y + z)$ for any real numbers $x$, $y$, and $z$, confirming the associative property of addition.

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Do you want more details, or do you have any questions? Here are some related questions for further exploration:

1. How would the proof change for multiplication?
2. Can the associative property be applied to subtraction? Why or why not?
3. Is the associative property valid in matrix addition?
4. How does the associative property differ from the commutative property?
5. How does the associative property relate to the distributive property?

Tip: The associative property only involves how terms are grouped, not the order of the terms. This differs from the commutative property, which involves changing the order of the terms.