Easy Algebra

1. Expand the square root.
   $\sqrt{\frac{x+xy}{y}-\frac{2y}{y}}$
   $=\sqrt{\frac{x+xy}{y}-2}$
   
   
2. The given expression is shown below.
   $x+\frac{x}{y}=83$
   $xy+x=83y$
   $\frac{xy+x}{y}=83$
   
3. Substitute.
   $\sqrt{\frac{x+xy}{y}-2}$
   $=\sqrt{83-2}$
   $=\sqrt{81}$
   $=\boxed{9}$

Since $x+\frac{x}{y}=83$ , we have

$\sqrt{\frac{x+xy-2y}{y}}$

$=\sqrt{\frac{x}{y}+\frac{xy}{y}-\frac{2y}{y}}$

$=\sqrt{\frac{x}{y}+x-2}$

$=\sqrt{83-2}$

$=\sqrt{81}$

$=\boxed{9},$

and we are done.

$x + \dfrac {x}{y} = 83$

$\dfrac {xy + x}{y} = 83$

$xy + x = 83y$

Inputting it into the square root, we get

$\sqrt {\dfrac {83y - 2y}{y}}$

$\sqrt {\dfrac {81y}{y}}$

$\sqrt {81}$

$9$

$\sqrt{\frac{x}{y}+\frac{xy}{y}-\frac{2y}{y}}$ since, $x+\frac{x}{y}=83,$ = $\sqrt{(83-x)+x-2}$ = $\sqrt{81}$ = 9

x+x/y=83 or, (x+xy)/y=83 or, (x+xy)/2y=83/2 [dividing both sides by 2] or, (x+xy-2y)/2y=(83-2)/2
or, (x+xy-2y)/y=81 [multiplying both sides by 2] sqrt((x+xy-2y)/y)=sqrt(81)=9 [ans]

$Given\quad That\\ x+\frac { x }{ y } =83\\ Now\quad Find,\quad \sqrt { \frac { x+xy-2y }{ y } } \\ =\sqrt { \frac { x }{ y } +\frac { xy }{ y } -\frac { 2y }{ y } } =\sqrt { x+\frac { x }{ y } -2 } \\ =\sqrt { 83-2 } \quad \quad \quad \quad \quad \quad \quad \quad :As\quad x+\frac { x }{ y } =83:\\ =\sqrt { 81 } =\boxed { 9 }$

$\sqrt{\frac{x+xy-2y}{y}} = \sqrt{\frac{x}{y}+x-2} = \sqrt{83-2} = \fbox{9}$

From given equation we get x+xy =83 y,by substituting this valoue in underroot values we get underroot 81 that is 9 Ans. K.K. GARG,India