# Thread: Any short method of solving these two equation

1. ## Any short method of solving these two equation

Plz solve this equation in a shortest way
X+root Y=7
root X+Y+11

2. Originally Posted by sandeeptiwari674
Plz solve this equation in a shortest way
X+root Y=7
root X+Y+11
First, this doesn't really belong in "Linear and Abstract Algebra".

I would solve $\displaystyle x+ \sqrt{y}= 7$ for x: $\displaystyle x= 7- \sqrt{y}$.

Then replace x by that in the second equation:
$\displaystyle \sqrt{7- \sqrt{y}}+ y= 11$
(You have "+11" but without an "=" there is no equation so I assume you mean "=".)

Now isolate the square root, $\displaystyle \sqrt{7- \sqrt{y}= 11- y$, and square both sides: $\displaystyle 7- \sqrt{y}= 121- 22y+ y^2$.

Again, isolate the square root, $\displaystyle -\sqrt{y}= y^2- 22y+ 114$ and square both sides.
$\displaystyle y= y^4- 22y^3+ 114y- 22y^3+ 484y^2+ 2508y+ 114y^2- 2508y+ 196$
$\displaystyle y= y^4- 44y^3+ 598y^2+ 484y+ 196$ or
$\displaystyle y^4- 44y^3+ 598y^2+ 483y+ 196= 0$.

That is a fourth degree polynomial equaton which might be difficult to solve. The only possible rational roots are the factors of 196 so I would recommend trying those first.

3. Originally Posted by sandeeptiwari674
Plz solve this equation in a shortest way
$\displaystyle X+\sqrt Y=7$
$\displaystyle \sqrt X+Y=11$
I couldn't help noticing that
$\displaystyle 4+\sqrt 9=7$
$\displaystyle \sqrt 4+9=11.$
Try following HallsofIvy's method, checking at each step whether those numbers fit the equations. That should lead to a solution.