Question

x²+xy+x=14,y²+xy+y=28,Then find tge value of x+y.Please give correct answer

Answer 1

x²+xy +x =14⇒x(x+y+1) =14

⇒x+y+1 = 14/x ---(1)
y²+xy+y =28⇒y(y+x+1) =28
⇒x+y+1 =28/y----(2)
therefore
(2) = (1)
28/y = 14/x
2/y =1/x
do the cross multiplication
2x = y ---(3)
put y= 2x in (1)
x+2x +1 =14/x
3x+1 =14/x

x(3x+1)=14
3x²+x-14=0
3x²+7x-6x -14=0
x(3x+7)-2(3x+7)=0
(3x+7)(x-2)=0
therefore
3x+7 =0 or x-2 =0
take x-2 =0
x=2
put x=2 in (3)
y=2x
y=2*2=4
therfore
x+y =2+4 =6

Answer 2

There are two answers possible:  -7 or 6.

x² + x y + x = 14
y² + x y + y = 28

Subtract both eq:   y² - x² + y - x = 14
                              (y - x) (y + x + 1) = 14    --- (3)

Add both eq:  (x + y)² + (x+y) = 42
                      (x+y) (x+y+1) = 42            --- (4)

(4) / (3)   gives,  (x+y+1 ≠ 0)
                      (x+y) / (y - x) = 3
         so          2 y = 4 x
                         y = 2 x

Substitute in (1) to get:  x² + 2 x² + x = 14
                                       x = [-1 +- √(1 + 168) ]/6 = -7/3  or  2

Then y = -14/3  or  4

so  x+y = 3 x = -7  or 6