Not that fast!

Algebra Level 2

The real numbers $x$ and $y$ are such that $x + \frac{2}{y} = \frac{8}{3}$ , and $y + \frac{2}{x} = 3$ .

The value $x + y$ can be expressed as $\frac{m}{n}$ for positive coprime integers $m$ and $n$ .

Find $m^{2} - n^{2}$ .

2 solutions

Arpit Sah
May 8, 2014

x+ $\frac{2}{y}$ = $\frac{8}{3}$

$\frac{2}{y}$ = $\frac{8}{3}$ - x

$\frac{2}{y}$ = $\frac{8 - 3x}{3}$

y = $\frac{6}{8 - 3x}$ ................. (1)

Substituting value of y in [ y + $\frac{2}{x}$ = 3 ] and after simplifying little we get :

$\frac{16}{8x - 3x^2}$ = 3

16 = 24x - $9x^2$

$9x^2$ - 24x + 16 = 0

$(3x - 4)^2$ = 0

3x - 4 = 0

x = $\frac{4}{3}$

Substituting value of x in (1) , we get:

y = $\frac{3}{2}$

Therefore, (x + y ) = $\frac{17}{6}$ = $\frac{m}{n}$

m = 17 and n = 6

$m^2$ - $n^2$ = 289 - 36 = $\boxed{253}$

Hazim Afifi
Apr 3, 2014

i used calculator....is there any way to solve this question without using calculator? -_-