Write a conjuecture that relates the result of the process to the original number selected

Given the process:

1. Add 7 to the original number \( x \).
2. Double the result.
3. Subtract 10.
4. Divide by 2.
5. Subtract the original number \( x \).

### Conjecture
For any original number \( x \), the result of this process will always be **2**, regardless of the value of \( x \).

### Proof:
Let's represent the process algebraically:

1. Start with the original number \( x \).
2. Add 7: \( x + 7 \).
3. Double the result: \( 2(x + 7) = 2x + 14 \).
4. Subtract 10: \( 2x + 14 - 10 = 2x + 4 \).
5. Divide by 2: \( \frac{2x + 4}{2} = x + 2 \).
6. Subtract the original number \( x \): \( (x + 2) - x = 2 \).

This confirms that the final result is always 2, regardless of the original number \( x \).

Would you like to explore this further or have any other questions?

Here are 5 related questions to deepen your understanding:
1. Can you apply this process to a non-integer number, and does the conjecture still hold?
2. How would you prove this conjecture using a different method, such as induction?
3. What other processes can be created that yield a constant result for any initial value?
4. How might the process change if you alter one of the operations, such as adding a different constant instead of 7?
5. What happens if you apply this process repeatedly—does the result remain constant, or does it change?

**Tip:** When forming conjectures, it's essential to test them with various inputs and, if possible, use algebraic proof to confirm their validity.

Explore a mathematical conjecture where a specific process always results in 2, regardless of the initial number. This problem involves arithmetic operations, algebraic expressions, and deductive reasoning. Suitable for Grades 6-8 students and those interested in mathematical patterns.

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