MHB Calculating Real $(x,y,z)$ with System of Equations

AI Thread Summary
The discussion focuses on solving a system of equations involving real numbers \(x\), \(y\), and \(z\) with their greatest integer and fractional parts. The initial equations are \(x[x]+z\{z\}-y\{y\} = 0.16\), \(y[y]+x\{x\}-z\{z\} = 0.25\), and \(z[z]+y\{y\}-x\{x\} = 0.49\). A participant suggests summing the equations to simplify the problem, resulting in \(x[x]+y[y]+z[z] = 0.9\). Another user advises eliminating the greatest integer and fractional parts by using the identity \(x^2 = x[x] + x\{x\}\) and recommends manipulating the equations further, such as subtracting one from another. The conversation emphasizes the need for strategic algebraic manipulation to solve for the variables.
juantheron
Messages
243
Reaction score
1
Calculation of Real $(x,y,z)$ in

$x[x]+z\{z\}-y\{y\} = 0.16$

$y[y]+x\{x\}-z\{z\} = 0.25$

$z[z]+y\{y\}-x\{x\} = 0.49$

where $[x] = $ Greatest Integer of $x$ and $\{x\} = $ fractional part of x

My try:: Add $(i) + (ii)+(iii)$

$x[x]+y[y]+z[z] = 0.9$

Now I did not Understand How Can I proceed after that,

please Help me

Thanks
 
Mathematics news on Phys.org
jacks said:
Calculation of Real $(x,y,z)$ in

$x[x]+z\{z\}-y\{y\} = 0.16$

$y[y]+x\{x\}-z\{z\} = 0.25$

$z[z]+y\{y\}-x\{x\} = 0.49$

where $[x] = $ Greatest Integer of $x$ and $\{x\} = $ fractional part of x

My try:: Add $(i) + (ii)+(iii)$

$x[x]+y[y]+z[z] = 0.9$

Now I did not Understand How Can I proceed after that,

please Help me

Thanks

Hi jacks! :)

It seems to me that you need to get rid of all those "greatest integer" and "fractional part" thingies.
What draws my attention is that they only occur in conjunction with the same variable.
Perhaps it is useful to consider that $x^2=x([x]+\{x\})=x[x]+x\{x\}$?
You might for instance subtract (iii) from (i) and apply that...
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.

Similar threads

Replies
1
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
4
Views
2K
Replies
3
Views
1K
Replies
8
Views
393
Back
Top