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I was told that when undoing operations in an equation you should start by following the PEDMAS rules, but in reverse. So, for example:

8x + 16/x = 4x + 16

According to the advice, I should start this problem by subtracting 16/x, since addition is the first operation to perform if your following PEDMAS backwards. However, online solvers of this problem perform multiplication first (by multiplying everything on both sides by x) in order to undo the division. I have seen similar problems solved in the same way, with the undoing of division being the first operation. My question is when is it your suppose to apply the reverse PEDMAS rule? Or does that rule really matter?

Jyrki Lahtonen
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Jason
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    IMO, there is no such rule to be followed strictly... In order to "solve" and expression you have to perform suitable operations in order to transform it into $ax=b$ , from which $x = \dfrac b a$. – Mauro ALLEGRANZA Apr 13 '19 at 13:20
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    This is the reason of the first step above: multiply both sides by $x$, in order to remove the $x$ in $\dfrac {16}{x}$. – Mauro ALLEGRANZA Apr 13 '19 at 13:20
  • What about an equation like: 2x + 2 = 8. In this equation you would get a different answer if you tried dividing the 2x first instead of subtracting the 2. Is it just that you need to recognize the certain equations where the reverse PEDMAS rule needs to be implemented? – Jason Apr 13 '19 at 13:46
  • Why different answers ? First subtract $2$ from both sides to get $2x=6$ i.e. $x=3$. Versus divide by $2$ both sides : $\dfrac {2x+2}{2}=\dfrac 8 2$ i.e. $x+1=4$. – Mauro ALLEGRANZA Apr 13 '19 at 14:06
  • The rules for solving equation are simply suitable application of the rules for equality : Substitution for functions "if $a=b$, then $f(a)=f(b)$". – Mauro ALLEGRANZA Apr 13 '19 at 14:10
  • Oh I wasn't dividing the 2 by 2 as well. The main reason I have this confusion is from this video on youtube: https://www.youtube.com/watch?v=LDIiYKYvvdA&list=PLUPEBWbAHUszT_GebJK23JHdd_Bss1N-G&index=4 talking about reverse PEDMAS and how it can make a big difference to solving an equation. – Jason Apr 13 '19 at 14:13
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    See the post Rules for transforming equations. – Mauro ALLEGRANZA Apr 13 '19 at 14:15
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    There is only one rule : if $A=B$, then $A+a=B+a$ and $A \times a = B \times a$. Basically you have only to apply it (with suitable choice of $a$). – Mauro ALLEGRANZA Apr 13 '19 at 14:18
  • Okay, well that gets to the heart of the matter. Thank you Mauro – Jason Apr 13 '19 at 14:21
  • Think of additions and subtractions as masking tape, multiplications and divisions as craft glue, and exponents as superglue. It's just how we stick numbers together (and pull them apart). Maybe parentheses represents a custom adhesive, even stronger than superglue. Epoxy Parentheses, now that has a ring to it. One final comment, the distributive property is apart and separate from the "order of operations". – John Joy Apr 13 '19 at 14:24

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Subtraction first makes sense, but not for the reason you think.

Use subtraction first to collect terms on one side and simplify, so: $$8x+\frac{16}{x}-4x-16=0$$ $$\to 4x-16+\frac{16}{x}=0$$ Then multiply to undo the division, so: $$4x^2-16x+16=0$$

Then we can solve by factorising.

Rhys Hughes
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