2011/073) Solve the system of quadratic equations

5x^2 + 4xy + 5y^2 + 3x + 3y = 74
x^2 + 2xy + y^2 - 6x - 6y = -8

5x^2 + 4xy + 5y^2 + 3x + 3y = 74

the coefficent of x^ and y^2 are same and so is of x and y

so we combine accordingly

5(x+y)^2 - 6xy + 3(x+y) = 74 ... 1


from 2nd equation by same anology

(x+y)^2 - 6(x+y) = -8

putting x + y = t

t^2 - 6t +8 = 0 or (t-4)(t-2) = 0 so t = 4 or 2

5(x+y)^2 - 6xy + 3(x+y) = 74
80-6xy + 12 = 74
or 6xy = 18
or xy - 3

x+ y= 4 .. 2
xy = 3 ... 3

from 2 and 3 we get (x-y)^2 = (x+y)^2 - 4xy = 16-12 = 4 or x-y = 2 or -2

x+y = 4 and x- y = 2 give x = 3 y = 1
x+y = 4 x-y = -2 give x = 1 and y = 3

putting x+ y =2 in 1 we get
5(x+y)^2 - 6xy + 3(x+y) = 74
20 - 6xy + 6 = 74
6xy = 48
xy = -8

so x = 4 y = -2 or x= -2 y = 4 (we can clove by above method)

so we get
(1,3), (3,1)(4,-2),(-2,4) 4 set of solutions