# [Solved] When multiplying both sides of an equation does not lead to a solution

#### ci139

Joined Jul 11, 2016
1,873
????? in common the way i like to see it – the equation of the form : $$x=0·y$$ – states that the variable x is not dependent on the variable y . . . or in other words the x is not the function of the variable y , nor vice versa .

#### WBahn

Joined Mar 31, 2012
28,163
????? in common the way i like to see it – the equation of the form : $$x=0·y$$ – states that the variable x is not dependent on the variable y . . . or in other words the x is not the function of the variable y , nor vice versa .
The discussion isn't about two different variables. But rather finding solutions to equations of a single variable.

f(x) = 0

when f(x) = g(x)/h(x)

The issue are the hidden gotchas involved with just multiplying both sides by h(x) and then solving g(x) = 0. This approach can erroneously find solutions that are incorrect since if h(x) = 0 at any point where g(x) is zero, the result is indeterminate. Further, it can miss solutions since f(x) = 0 also occurs at any x where g(x) is non-zero but finite if h(x) becomes infinite.

#### MrAl

Joined Jun 17, 2014
9,761
The discussion isn't about two different variables. But rather finding solutions to equations of a single variable.

f(x) = 0

when f(x) = g(x)/h(x)

The issue are the hidden gotchas involved with just multiplying both sides by h(x) and then solving g(x) = 0. This approach can erroneously find solutions that are incorrect since if h(x) = 0 at any point where g(x) is zero, the result is indeterminate. Further, it can miss solutions since f(x) = 0 also occurs at any x where g(x) is non-zero but finite if h(x) becomes infinite.
Hello again,

I think he started with x/y=0 where y is just the denominator terms.

But i think you may be putting too much emphasis on how 'bad' multiplying by 'y' is here in:
x/y=0
x=0*y

There are times when it definitely works and there are times when this idea is needed i think.
For a simple example, solve x/y=0. If you cant multiply by 'y' then you cant solve the general x/K=0 either and that can certainly come up in various situations. So there needs to be a balance when discussing this idea. It's good to know that there are times when there will be solutions we cant find like this, but it's also good to know that we can find solutions like this, and that sometimes the solutions we miss are not practical anyway. An example is:
sqrt(a)=2
and here we know that two solutions are a=4 and a=-4, but in many instances the solution -4 is not usable simply because it is negative and things like negative resistors are not usually what we are looking for.
Also, the simple sin(x)/x=0 is easy to examine without doing any work, but when the solutions are not that obvious you cant tell what solutions are usable and which are not.
For a real life example here is an equation:

(4*a^2*f*pi-8*f*pi*(b-4*f^2*pi^2))/(2*sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*sqrt((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2))-
(sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*(4*f*pi*A^2-8*f*pi*(B-4*f^2*pi^2)))/(2*((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2)^(3/2))=0

Solve that for the center frequency f.
This isnt too hard to solve if you are "allowed" to multiply both sides by the denominator, and you do in fact come up with the useful solution(s), and even though they are usable in some applications only one of them is useful in other applications or theory.
If you really want to you can try to find other solutions using that equation however you wish i dont think any of them will be useful.

The real point though is that you are making it sound like we can never ever do this when sometimes we certainly can and it's good to do that. I do agree that it is wise to think about it though and realize we may miss some solutions that are relevant to some applications. Qualifiers like "dangerous" and "harmful" are not the right words i believe and are misleading. I think the better word would be "caution".

#### WBahn

Joined Mar 31, 2012
28,163
Notice that I have NEVER said that you can't ever multiply both sides by the denominator. I have said that you should never do it BLINDLY. You should always ask whether the denominator can either go to zero or infinity for any value of the variable within the range of interest. If it can't, then you are good to go; but if it can, then be sure to deal with those cases appropriately.

#### MrAl

Joined Jun 17, 2014
9,761
Notice that I have NEVER said that you can't ever multiply both sides by the denominator. I have said that you should never do it BLINDLY. You should always ask whether the denominator can either go to zero or infinity for any value of the variable within the range of interest. If it can't, then you are good to go; but if it can, then be sure to deal with those cases appropriately.

Well consider this again:
(4*a^2*f*pi-8*f*pi*(b-4*f^2*pi^2))/(2*sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*sqrt((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2))-
(sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*(4*f*pi*A^2-8*f*pi*(B-4*f^2*pi^2)))/(2*((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2)^(3/2))=0
and to solve this factor then multiply both sides by the denominator of the left side.

And then the opening statement:
"In general, this is dangerous".

So it ends up sounding like the technique is being discouraged completely right off the bat.
Now if the original statement was:
"Stand as close as possible to the edge of the cliff"
"In general this is dangerous"
would make a LOT of sense

It's nice that you recognize the fact that it is actually useful though.

#### WBahn

Joined Mar 31, 2012
28,163
Well consider this again:
(4*a^2*f*pi-8*f*pi*(b-4*f^2*pi^2))/(2*sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*sqrt((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2))-
(sqrt((b-4*f^2*pi^2)^2+4*a^2*f^2*pi^2)*(4*f*pi*A^2-8*f*pi*(B-4*f^2*pi^2)))/(2*((B-4*f^2*pi^2)^2+4*f^2*pi^2*A^2)^(3/2))=0
and to solve this factor then multiply both sides by the denominator of the left side.

And then the opening statement:
"In general, this is dangerous".

So it ends up sounding like the technique is being discouraged completely right off the bat.
Now if the original statement was:
"Stand as close as possible to the edge of the cliff"
"In general this is dangerous"
would make a LOT of sense

It's nice that you recognize the fact that it is actually useful though.
In general, the approach you suggested is dangerous because the approach you suggested was to blindly get rid of the denominator in order to simplify the equation.

Please read the very next sentence that explains why that approach can be dangerous and the rest of the post explained how to remove that danger.

#### MrAl

Joined Jun 17, 2014
9,761
In general, the approach you suggested is dangerous because the approach you suggested was to blindly get rid of the denominator in order to simplify the equation.

Please read the very next sentence that explains why that approach can be dangerous and the rest of the post explained how to remove that danger.
Hello again,

Ok good luck with it.