So here I was, making 2 math problems, I was able to solve them, but 2 operations seem a bit intractable to me. Maybe you can help me understand why this is true:
The first problem:
$$x = \frac{1}{5} - \frac{4}{y}$$
$$\frac{4}{y} =\frac{1}{5} - x$$
$$\frac{4}{y} = \frac{1-5x}{5} $$
$$\frac{y}{4} = \frac{5}{1-5x}$$
Why is it possible to turn $\frac{y}{4}$ upside down?
$$y = 20 / 1 - 5 x$$
The second problem:
$$4A√B - √B = 3$$
$$√B(4A-1) = 3$$
Where does the -√B go? I understand that the -1 comes from the - sign in front of the square root. But where does the other √B go?
$$\sqrt{B} = \frac{3}{(4A-1)}$$
$$B = (\frac{3}{(4A-1)} )^2$$
$$B = \frac{9}{(4A-1)^2}$$
Everything, except above the bold text I understand.
Maybe I do not understand the full extent of a certain rule which I am familiar with in simpler situations. That's why I think an example would be very useful. I really want to have a deep understand of why these things are true.
Greetings,
Bowser.
Answer
First question: if two fractions are equal, then their reciprocals are equal too (the reciprocal of a fraction is the same fraction "turned upside down", using your terminology).
Second question: the $B$ is still there: if you multiply $\sqrt{B}(4A-1)$ you get indeed $4A\sqrt{B}-\sqrt{B}$. It's called "distributive law".
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