How to find variable equation solution

Question

My adopted method to find solution

\begin{align} &\frac{-4x + 40}{x-7} &=& \frac{4x-40}{13 - x}\\ \implies& \frac{-(4x - 40)}{x-7} &=& \frac{4x - 40}{13 - x}\\ \implies& \frac{-(4x - 40)}{x-7}\times \frac{1}{4x - 40} &=& \frac{4x - 40}{13 - x} \times \frac{1}{4x - 40}\\ \implies& \frac{-1}{x - 7} &=& \frac{1}{13 - x} \\ \implies& \frac{-1}{x-7} &=& \frac{1}{13 - x}\\ \implies& x - 13 &=& x - 7 \end{align}

Please tell me what basic mathematics rule I violated to get such a wrong answer . PS I know how the right solution , I just need to know what is wrong with my current method .

Answer 1

You have made one mistake:

• You have divided both sides by $4x-40$ but you first have to check if $4x-40$ is equal to $0$ or not. So first let take $4x-40=0 \implies x=10$. We notice $x=10$ is fulfiling the original equation.

Special point:

• You have got $\dfrac{-1}{x-7}=\dfrac{1}{13-x} \implies x-13=x-7$. Since this equation has no possible solution the only solution is the first point answer or $x=10$