Mathematical teaser equation

**Coach** · August 17th 2008, 07:03 AM

Hello everyone!

I hope everyone has had a relaxing summer.

Anyway, I have ventured again into a realm of math that is far beyond my abilities, in other words I am trying to solve some mathematical problems from the book Challenging Mathematical Teasers by J.A.H. Hunter

The word problem I am currently solving leads to the following two equations

$<br /> x^2-y^2=z^3<br />$

and
$\frac{x}{y}=\frac{y}{z}$

by substiuting we will have $x^2-xz=z^3$

This is about as far as I got, but then the solutions says something that I can't quite understand.

This equation is fully satisfied by: $x-z=mz$ , and x= $\frac{z^2}{m}$ , where m is any rational number.

Could someone pleade exmplain the above thing to me? Does the author of this book use some systematic approach to solving these kind of equations with 3 unknowns or is it just the mathematician's creativity?

All responses are appreciated.

**CaptainBlack** · August 17th 2008, 07:35 AM

Originally Posted by Coach

Hello everyone!

I hope everyone has had a relaxing summer.

Anyway, I have ventured again into a realm of math that is far beyond my abilities, in other words I am trying to solve some mathematical problems from the book Challenging Mathematical Teasers by J.A.H. Hunter

The word problem I am currently solving leads to the following two equations

$<br /> x^2-y^2=z^3<br />$

and
$\frac{x}{y}=\frac{y}{z}$

by substiuting we will have $x^2-xz=z^3$

This is about as far as I got, but then the solutions says something that I can't quite understand.

This equation is fully satisfied by: $x-z=mz$ , and x= $\frac{z^2}{m}$ , where m is any rational numberd.

This is just a change of variable, what ever the value of $x$ and $z$ we can write $x-z=mz$ for some $m$ , then because $x(x-z)=z^3$ we have of necessity that $x=z^2/m$ . The advantage is now that by equating the two expressions for $x$ you have a quadratic relating $m$ and $z$ , so you can solve this for $m$ in terms of $z$ , and so the solutions can be written in tems of a single parameter $z$ .

(note you don't need to assume that $m$ is rational, at least I can't see why its necessary unless we are looking for integer or rational solutions)

RonL

**vishalgarg** · August 22nd 2008, 12:41 AM

Coach you've got the equatin
x^2 - xz = z^3
or

x(x-z) = z^3
x-z = z^3/x
now put x= z^2/m

x-z = zm...
so you got it.

**Lucy.Gray** · August 25th 2008, 12:45 AM

May i ask a question about "Depreciation"?

**Jhevon** · August 25th 2008, 08:25 AM

Originally Posted by Lucy.Gray

May i ask a question about "Depreciation"?

of course! you are more than welcome to... in a new thread

Thread: Mathematical teaser equation

LinkBack

Thread Tools

Display

Mathematical teaser equation

Similar Math Help Forum Discussions

mathematical equation

Proof by mathematical induction of a recursion equation

Brain Teaser-PLS HELP!

Brain Teaser Help Please~!

Help... math teaser

Search Tags