Algebraic Solution of System of Equation involving quadratic

Question

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

9x^2 + 18xy - 6y^2 = 3
Find x and y

Answer 1

3x^2=2-4xy-3y^2                9x^2=3-18xy-6y^2

                                           (9x^2)/3=(3-18xy-6y^2)/3

                                            3x^2=1-6xy-2y^2

now the 2 fuctions are equal 3X^2 so the other side its equal too.

 

2-4xy-3y^2=1-6xy-2y^2

 

1+2xy-y^2=0    *-1

 

y^2-2xy-1=0

x=(1-y^2)/y

 

and you need substitute de x in the firsts equations and do all the transformations, but that part Its all yours.

 

 

Answer 2

These appear to be two rotated conic sections. Two methods are possible.

 

1.

Using the quadratic formula to express y in terms of x, and substituting in to the other equation.

 

2. 

Rotating the axis (cos(2θ) = (A - C) / B), determining the common solutions, then rotating the axis back.

 

Both of these possibilities are very time intensive, not to mention the possibility of arithmetic mistakes. 

Personally, I do not think the solution is worth the time and effort.