Question

Пожалуйста помогите Найдите ошибку в доказательстве: "Пусть а=b+с , тогда а(а-b)= (b+c)(a-b), следовательно, а²-аb=аb+ас-b²-bс, затем а²-аb-ac=ab-b²-bc и, наконец а(а-b-c)=b(a-b-c). Значит, a=b".

Answer 1

A-b-c =0, поскольку a=b+c.
Обе части тождества поделили на ноль, что недопустимо.

Answer 2

Error in the Proof

The given proof attempts to show that if \(a = b + c\), then \(a = b\). Let's analyze the steps of the proof to identify the error.

1. Initial Equation: \(a = b + c\)

2. Multiplying Both Sides by \(a - b\): - The proof states: \(a(a - b) = (b + c)(a - b)\)

3. Expanding Both Sides: - The proof states: \(a^2 - ab = ab + ac - b^2 - bc\)

4. Rearranging Terms: - The proof states: \(a^2 - ab - ac = ab - b^2 - bc\)

5. Factoring Out \(a - b - c\): - The proof states: \(a(a - b - c) = b(a - b - c)\)

6. Conclusion: The proof concludes that \(a = b\).

Error Analysis

The error in the proof lies in the step where the terms are rearranged. Let's analyze this step in detail.

The step where the terms are rearranged: \[a^2 - ab - ac = ab - b^2 - bc\]

The error occurs when rearranging the terms. The correct rearrangement should be: \[a^2 - ab - ac + b^2 + bc = 0\]

Conclusion

The error in the given proof lies in the incorrect rearrangement of terms. Therefore, the conclusion that \(a = b\) is not valid based on the provided proof.