My Math Forum algebra to invert fractional equation?

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 December 8th, 2014, 08:02 AM #1 Newbie   Joined: Mar 2013 Posts: 9 Thanks: 0 algebra to invert fractional equation? so.. if a/b=c/d then b/a=d/c because you can use each side of the equation do divide one by to invert. how do i represent this as an explicit step in place of just flipping the fractions? would a/b=c/d 1/(a/b)=1/(c/d) b/a=d/c be legit algebra? this seems wrong somehow.
 December 8th, 2014, 08:18 AM #2 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,142 Thanks: 726 Math Focus: Physics, mathematical modelling, numerical and computational solutions Yep. Doing "1 over" is a valid mathematics operation on both sides of an equation provided you don't have any zeroes. It's okay to do $\displaystyle x = -y$ $\displaystyle \frac{1}{x} = -\frac{1}{y}$ (if x and y are non-zero), but not $\displaystyle x+y = 0$ $\displaystyle \frac{1}{x+y} = \frac{1}{0}$ because 1 divided by 0 is undefined. Also, you need to pay attention to the fact that in the first case presented above, x and y must be non-zero. Otherwise you're doing the same thing as the second case. As for your specific example, another way of showing that they are the same is to multiply both sides by $\displaystyle \frac{bd}{ac}$ $\displaystyle \frac{a}{b} = \frac{c}{d}$ $\displaystyle \frac{a}{b} \times \frac{bd}{ac}= \frac{c}{d} \times \frac{bd}{ac}$ $\displaystyle \frac{\cancel{a}}{\cancel{b}} \times \frac{\cancel{b}d}{\cancel{a}c}= \frac{\cancel{c}}{\cancel{d}} \times \frac{b\cancel{d}}{a\cancel{c}}$ $\displaystyle \frac{d}{c} = \frac{b}{a}$ which is okay if a, b, c and d are non-zero.
 December 8th, 2014, 10:32 AM #3 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Another way of simplifying 1/(a/b) with a*b is nonzero, is as follows: $\displaystyle \frac{1}{\frac{a}{b}} = \frac{1}{\frac{a}{b}} \cdot 1 = \frac{1}{\frac{a}{b}} \cdot \frac{b}{b} = \frac{b}{\frac{a}{b} \cdot b} = \frac{b}{\frac{ab}{b}} = \frac{b}{ \frac{b}{b} \cdot a} = \frac{b}{1 \cdot a} = \frac{b}{a}$ where you might like to leave out some steps.

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