The discussion isn't about two different variables. But rather finding solutions to equations of a single variable.?????
in commonthe 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 .
Hello again,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.
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.
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.Well consider this again:
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"
then the reply:
"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.
Hello again,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.
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