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In a YouTube video by Michael Penn, at the 3:56 mark, he gets the equation: $$ \frac{4}{2} = \frac{2}{2-x}\Rightarrow 8-4x=4$$ Could someone explain to me how he does this and the process behind it? Thanks for your help! (Im only freshman year of high school so sorry if this question seems basic)

Alexander Benton
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  • Cross-multiplying: if $\dfrac ab=\dfrac cd$, then (multiply both sides by $bd$ to get) $ad=bc$ – J. W. Tanner Nov 02 '20 at 18:43
  • Wouldn't the observation $$ \frac{4}{2} = \frac{2}{2-x}\iff \frac{2}{1} = \frac{2}{2-x}$$ lead immediately to the conclusion $1=2-x$? – Michael Hoppe Nov 03 '20 at 12:28
  • Could those voting the close please say what their problem with this question is? This is a reasonable question for someone at this level. – user1729 Nov 13 '20 at 10:37

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The domain is $x\neq2$ and we obtain: $$\frac{4}{2}\cdot2(2-x)=\frac{2}{2-x}\cdot2(2-x),$$ which gives $$4(2-x)=4$$ or $$8-4x=4.$$ Here $2(2-x)$ it's a common denominator of the left side and the right side of the equation.

Michael Rozenberg
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  • +1 elementary yet elegant answer. BTW, do you have an email to correspond in research? –  Jan 17 '21 at 05:59