Can this be solved with any HP CAS?

[Csaba Tizedes]

(09-29-2017 02:26 AM)SlideRule Wrote:  ...the expression a+b = c+b represents the original equation...

Unfortunately this is not true, because we have no information about 'b' when we substract it from both sides of the equation.

The correct way, as we learned in the school:

In \(\frac{1}{x-5}\) the \(x-5\neq0\), therefore \(x\neq5\), so \( x\in\Bbb R\backslash\{5\} \)

If \(x\neq5\), \(\frac{1}{x-5}\) is a number, therefore we can substract both sides of equation and we can solve the equation \(3x=15\), the solution is \(x=5\). But \( x\in\Bbb R\backslash\{5\} \), therefore \(x=5\) is not a solution.

The right solution of the original equation is "no solution".


The Skynet: If Skynet has a good CAS that will never found \(x=5\), because that CAS know that \( x\in\Bbb R\backslash\{5\} \). But the CAS of any HP, TI, Maple, Mathematica, etc... gives the wrong \(x=5\) - so the world will be destroyed...


Csaba

[SlideRule]

(09-29-2017 05:57 PM)Csaba Tizedes Wrote:  .. this is not true, because we have no information about 'b' when we substract it from both sides of the equation ... \(x=5\) is not a solution. The right solution of the original equation is "no solution".

OK, then expand the equation, collect terms & then simplify.

3x+(1/(x-5)) = 15+(1/(x-5))
expands to
3x² - 15x + 1 = 15x - 75 + 1
collect terms
3x² - 30x + 75 = 0
factor
3(x² - 10x + 25) = 0
factor
3(x-5)² = 0
so either
3=0 or (x-5)=0
since 3 ≠ 0
then
x-5 = 0
solve for x
x = 5

BEST!
SlideRule

ps: added Wolfram Alpha output(s)
[attachment=5208]

[attachment=5207]

[lrdheat]

Is the term 1/(x-5) on both sides simply the additional restraint on the problem which would be the same as writing it as solve 3*x=15|x not =5 ? This is clearly undefined on the real number case where as factoring out the 1/(x-5) terms on both sides makes it clearly x=5

What was the context, the reason for the 1/(x-5) term in the problem?

[Csaba Tizedes]

(09-29-2017 10:32 PM)lrdheat Wrote:  What was the context, the reason for the 1/(x-5) term in the problem?

You're right, this is equivalent 3×X=15 and X!=5 problem.
This is an example from a hungarian middle school mathematics book.
The reason of the term 1/(X-5) to check the student can to reduce the set of the real numbers (5 is not belongs to \( \Bbb R\) )

I just use it to test CAS systems - and all of them fails.

I really do not understand why do you do not see, this is a really big problem. How you can trust a system, if that system cannot solve this simple, midschool straigh example.


Csaba

[Nigel (UK)]

(09-29-2017 06:44 PM)SlideRule Wrote:  ...
OK, then expand the equation, collect terms & then simplify.

3x+(1/(x-5)) = 15+(1/(x-5))
expands to
3x² - 15x + 1 = 15x - 75 + 1
collect terms
...
BEST!
SlideRule

The problem is that the second equation above is only equivalent to the original equation when \(x\neq5\), because going from the second equation to the original equation requires division by \((x-5)\) and when \(x=5\) this isn't allowed. Because of this a solution to the second equation may not be a solution to the first equation.

Disclaimer: I am not a mathematician! This is just my understanding of the situation.

Nigel (UK)

[SlideRule]

(09-30-2017 09:10 AM)Nigel (UK) Wrote:  ... is only equivalent to the original equation when \(x\neq5\) ...

EXACTLY! BUT how do you know this is true until you solve the expression? It's not so much that the expression can not be solved as the solution requires understanding. You have two polynomials set in an equivalency expression that appear to have been divided by an expression which is equivalent to zero WHEN the variable common to both expressions is set to 5 (one interpretation), where 5 is also the solution to the quadratic expression prior to a synthetic division. This is a useful process and analysis, wherein the results need interpretation. Wolfram does this, look at the attached Jpg's in my previous post.
The solution reveals the undefined operation.

BEST!
SlideRule

[Gilles59]

(09-28-2017 07:43 PM)Csaba Tizedes Wrote:  

(09-27-2017 08:56 PM)Gilles59 Wrote:  I suppose that the CAS simplify too much ;D

PS.: No any CAS which can solve this problem correctly... This is too simple to handle it correctly. Of course the triple integrals in negative dimensions on a hyperboloid surface can do without problem, but I'm an engineer and I want RELIABLE solutions.

Hi CSaba

The HP Prime returns the correct answer {}
The old hp50 cas simplify the équation in 3x=15. but we have a brain. A cas is just a tool. This limitation of DOMAIN for the 50g is explain in the doc.

You are engineer. Perhaps for your problem x=5+/- epsilon could be an interesting approximate solution. For an engineer the question is : what means x=5 in the system? Infinite length? Infinite force? Undeground altitude? Negative mass? Why x=5 is impossible in a physical point of view? For an engineer mathematic is just a tool in my point of view.

[Csaba Tizedes]

(09-30-2017 01:04 PM)Gilles59 Wrote:  but we have a brain. A cas is just a tool. This limitation of DOMAIN for the 50g is explain in the doc.

Yes, my biggest problem is that, the English is not my native and I cannot explain subtly what I want to say. Of course I dont want to underestimate anybody here.
I stopped at 48GX, but I have an 48GII - as I know that is same as 50G, and found X=5.

(09-30-2017 01:04 PM)Gilles59 Wrote:  You are engineer. Why x=5 is impossible in a physical point of view? For an engineer mathematic is just a tool in my point of view.

This is my luck. The real problems are so smooth and has derivative in any point

Csaba

[Gilles59]

(09-30-2017 07:16 PM)Csaba Tizedes Wrote:  

(09-30-2017 01:04 PM)Gilles59 Wrote:  but we have a brain. A cas is just a tool. This limitation of DOMAIN for the 50g is explain in the doc.

Yes, my biggest problem is that, the English is not my native and I cannot explain subtly what I want to say.Csaba

It's the same for me. There is a big difference between what I want to say and what i write in English ;D Your exemple is interesting about CAS limitations. I remember that my math teachers said so many time " hummm but what is the domain of definition of ....".

[SlideRule]

"When mathematics is used as a tool, it cannot be guided exclusively by internal mathematical reasoning. Instead, what qualifies as adequate tool-use is also determined by the problem at hand and its context. Consequently, tool-use has to respect conditions like suitability, efficacy, optimality, and others. Of course, no tool will provide anything like the unique solution. On the contrary, the notion of tool stresses that there is a spectrum of means that will normally differ in how well they serve particular purposes. This practical outlook demands a new view on the concept of validity in mathematics. The traditional philosophical stance emphasizes the permanent validity of mathematical theorems as a pivotal feature.
The tool perspective, in contrast, underlines the inevitably provisional validity of mathematics: any tool can be adjusted, improved, or lose its adequacy upon changing practical conditions". Introduction, pg 1-2
Mathematics as a Tool
ISBN 978-3-319-54468-7
ISBN 978-3-319-54469-4 (eBook)
DOI 10.1007/978-3-319-54469-4
© Springer 2017

BEST!
SlideRule