MHB Find Value of $x-y+z$ in $x,y,z \in N$

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Value
Albert1
Messages
1,221
Reaction score
0
$x,y,z \in N\\
\left\{\begin{matrix}
3x-4y=0---(1)\\
x+y+z=\sqrt {x+y+z-3}+15---(2)
\end{matrix}\right.$
find the value of $x-y+z=?$
 
Mathematics news on Phys.org
Albert said:
$x,y,z \in N\\
\left\{\begin{matrix}
3x-4y=0---(1)\\
x+y+z=\sqrt {x+y+z-3}+15---(2)
\end{matrix}\right.$
find the value of $x-y+z=?$

from 2nd relation we get $x+y+z=19$

from (1) and above

$ x = 4, y = 3, z = 12 $ giving $x-y+z = 13$

or
$x = 8, y = 6, z = 5$ giving $x-y+z = 7$
 
kaliprasad said:
from 2nd relation we get $x+y+z=19$
That's clever. The 2nd relation is $x+ y+ z= \sqrt{x+ y+ z- 3}+ 15$. Subtracting 3 from both sides, $x+ y+ z- 3= \sqrt{x+ y+ z- 3}+ 13$. Letting u= x+ y+ z- 3, We can write that as $u= \sqrt{u}+ 12$. And that is the same as $\sqrt{u}= u- 12$. Squaring both sides, $u= (u- 12)^2= u^2- 24u+ 144$ s that $u^2- 25u+ 144= (u- 16)(u- 9)= 0$. The roots of that are u= 16 and u= 9. If u= 16 then x+ y+ z= u+ 3= 19. If u= 9 the x+ y+ z= u+ 3= 12.

Having squared both sides we need to check for spurious solutions that might have been introduced. If x+ y+ z= 19, then the equation becomes $19= \sqrt{19- 3}+15= 4+ 15$ which is correct. If x+ y+ z= 12, then the equation becomes $12= \sqrt{12- 3}+ 15= 3+ 15$ which is not correct (we got this "spurious solution" because -3+ 15 is equal to 12.)

I'm impressed!

from (1) and above

$ x = 4, y = 3, z = 12 $ giving $x-y+z = 13$

or
$x = 8, y = 6, z = 5$ giving $x-y+z = 7$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top