Confusing Magic Sum

Question

When this equation is solved in this way

$$\frac{-(4x-40)}{x-7}= \frac{4x-40}{13 -x}$$
$$\frac{- 1}{x-7} =\frac{ 1}{ 13 -x}$$
$$-x + 7 = 13 - x$$
$$13 \neq 7$$

WHY THIS?

Answer 1

From $- (4x-40) / (x-7)= (4x-40) / (13 -x) $ to $- 1 / (x-7) = 1 / (13 -x) $ you have divided by $4x-40$. But this is not allowed if $x=10$.

If $x \ne 10$ you get $7=13$, which is absurd ! Hence the equation $- (4x-40) / (x-7)= (4x-40) / (13 -x) $ has only one solution: $x=10$.

Answer 2

You have indeed done “magic math” using a false rule of inference. The “rule” you have used is this:

$$ab = ac \implies b=c. $$

This “rule” is never valid, but many people have found it amusing to sneak this rule into fake “proofs” that $1=2$ or some other absurd result.

A correct rule that applies in your situation is

$$ ab = ac \implies b=c \ \ \text{OR}\ \ a =0.$$

Applying this rule with the substitution $4x-40$ for $a.$ $-1/(x-7)$ for $b,$ and $1/(13-x)$ for $c,$ we eventually reach the conclusion that either $7=13$ or $x=10.$ Since $7\neq 13$ the only possible true answer is that $x=10.$