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$\frac{52}{x}=13$

It says to next step

$\frac{52}{13}=x$

Ok, I can do future problems like this, but is there a rule that explains this? What just happened to both sides of the equal sign?

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5  
he preformed two operations simultaneously, consider he multiplied both sides by $x$, then after divided both sides by $13$. Often authors will not explicitly preform every step so keep an eye out for what valid operations you could preform at every stage of a problem. –  Deven Ware Sep 14 '11 at 3:21
    
if my comment helped realize that both answers now explain what I said in more detail so look at them carefully and understand them –  Deven Ware Sep 14 '11 at 3:25

3 Answers 3

up vote 7 down vote accepted

There are two steps here: starting with $$\frac{52}{x}=13$$ we multiply both sides by $x$ to get $$\frac{52}{x}\cdot x=13\cdot x$$ $$52=13\cdot x$$ and then dividing both sides of that by $13$ to get $$\frac{52}{13}=\frac{13\cdot x}{13}$$ $$\frac{52}{13}=x$$

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I'd count several steps here. Anyways...

We start with $$\left(\frac{52}{x}\right)=13$$. Multiplying both sides by x we have $$\left(\left(\frac{52}{x}\right)*{x}\right)=(13*{x}).$$ Since $$\left(\frac{x}{1}\right)=x$$ we have $$\left(\left(\frac{52}{x}\right)*\left(\frac{x}{1}\right)\right)=(13*{x}).$$ Applying the product rule for quotients, we have $$\left(\frac{(52*x)}{(x*1)}\right)=(13*{x}).$$ Since multiplication commutes, we have $$\left(\frac{(x*52)}{(x*1)}\right)=(13*{x}).$$ Applying the product rule for quotients again, we have $$\left(\left(\frac{x}{x}\right)*\left(\frac{52}{1}\right)\right)=(13*{x}).$$ Since $$\left(\frac{x}{x}\right) =\left(\frac{1}{1}\right)$$ we obtain $$\left(\left(\frac{1}{1}\right)*\left(\frac{52}{1}\right)\right)=(13*{x}).$$ Applying the product rule for quotients once again we have $$\left(\frac{(1*52)}{(1*1)}\right)=(13*{x}).$$ Since $(1*1)=1$, and $(1*52)=52$ we obtain $$\left(\frac{52}{1}\right)=(13*{x}).$$ Now we multiply both sides by $$\left(\frac{1}{13}\right)$$ and obtain $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\left(\frac{1}{13}\right)*\left(13*{x}\right)\right).$$ Since 13 equals $$\left(\frac{13}{1}\right)$$ we obtain $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\frac{1}{13}\right)*\left(\left(\frac{13}{1})*{x}\right)\right).$$ Since * associates we have $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\left(\left(\frac{1}{13}\right)*\left(\frac{13}{1}\right)\right)*{x}\right).$$ Applying the product rule for quotients on both sides we obtain $$\left(\frac{(1*52)}{(13*1)}\right)=\left(\left(\frac{(1*13)}{(13*1)}\right)*{x}\right).$$ Since $(1*52)=52$, $(13*1)=13$, and $(1*13)=13$, we obtain $$\left(\frac{52}{13}\right)=\left(\left(\frac{13}{13}\right)*{x}\right).$$ Since $$\left(\frac{13}{13}\right)=1$$ we obtain $$\left(\frac{52}{13}\right)=(1*x).$$ Since $(1*x)=x$, we obtain $$\left(\frac{52}{13}\right)=x$$

Even reasoning in this way steps have gotten left out, such as how we can "multiply both sides by x."

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Thanks for the edit Zev. From the downvotes it looks like some people don't like this sort of response even if it comes in "standard" notation. –  Doug Spoonwood Sep 14 '11 at 12:46

For $a, b, c, d$ non-zero,

$\frac{a}{b} = \frac{c}{d}$ is equivalent to

$ad = bc$.

Then if you divide both sides by $c$, you'll get $\frac{ad}{c} = b$

Believe it or not, this is your situation! (with $d = 1$)

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