Ytukyg

If we have 2 fractions and set them equal to each other, can we remove the numerator without changing the equation? So the demonators would just equal eachother, so $4p = 8$ Like this:

$$frac{1}{4p} = frac{1}{8}$$
Multiply each side by 4p:

$$frac{4p*1}{4p} = frac{4p*1}{8}$$
Then we cross out each 4p on the LHS:
$$1 = frac{4p}{8}$$
Then times each side by 8 to isolate p:
$$1*8 = frac{4p*8}{8}$$
And finnally we cross out each 8 on the LHS and left with:
$$8= 4p$$
Can we do this with any number? Assuming of course the number is real.

Suppose we have $$frac{a}{b}=frac{a}{c}$$

where $a ne 0$.

Multiplying both sides by $frac{bc}{a}$ gives us $c=b$.

Assuming $p neq 0 $ (which is necessary for LHS not being undefined), you can directly multiply by the denominators' Minimum Common Multiple to yield:

$$
frac{1}{4p}=frac{1}{8} quadbigg|cdot 32p iff 8=4p iff p=2$$

Another point of view:

The function $,,f(x):=frac1x, quad f: mathbb{R^*}longrightarrow mathbb{R^*},,$ is bijective (or one-to-one), which by definition means that:

$$textit{If}quad f(a)=f(b)quad textbf{then}quad a=b$$

Your equation is nothing else than

$$f(4p)=f(8)$$

Considering $f$'s bijective:

$$4p=8 iff p=2$$