How to convince a math teacher of this simple and obvious fact?

Question 1

I have in my presence a mathematics teacher, who asserts that

$\frac{a}{b} = \frac{c}{d}$

Implies:

$a = c, \space b=d$

She has been shown in multiple ways why this is not true:

$\frac{1}{2} = \frac{4}{8}$

$\frac{0}{5} = \frac{0}{657}$

For me, these seem like valid (dis)proofs by contradiction, but she isn’t satisfied. She wants a ‘more mathematical’ proof, and I can’t think of any.

I’m worried that if she isn’t convinced, it may be detrimental to some students. Is there another way to systematically demonstrate the untruth of her conjecture?

EDIT: Since the answer which worked was from a comment, but each answer is also very good, I’m upvoting all of them instead of accepting a specific one. Feel free to close this question for being too open if so a moderator desires.

Question 2

You can prove that all the numbers are equal 😉

Let’s assume that for all $a,b,c,d \in \mathbb{R}$ , $b \neq 0$ , $d \neq 0$ we have

$\frac{a}{b} = \frac{c}{d}\quad \text{ implies }\quad a = c\ \text{ and }\ b = d. \tag{$\spadesuit$}$

Now take any two numbers, say $p$ and $q$ , and write

$\frac{p}{p} = \frac{q}{q}.$

Using claim $(\spadesuit)$ we have $p = q$ . For the special case, where one of them equals zero (e.g. $q$ ), use $\frac{2p}{2p} = \frac{p+q}{p+q}.$

I hope this helps 😉

How to convince a math teacher of this simple and obvious fact?

Answer

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