x-7y=2 ​ x(sq)=34y(sq)+7xy-16

x - 7y  =  2          and         x²  =  34y² + 7xy  - 16

 

Take the linear equation (x - 7y  =  2) and solve it for either  x  or  y.

It seems easier to solve it for x:  x  =  7y + 2

 

Replace this into the other equation:

  x²  =  34y² + 7xy  - 16     --->       (7y + 2)²  =  34y² + 7(7y + 2)y  - 16

Simplify:     --->     49y² + 28y + 4  =  34y² + 49y² + 14y - 16

Simplify:     --->     0  =  34y² - 14y - 20

Divide both sides by 2:     0  =  17y² - 7y - 10

Factor:                             0  =  (17y + 10)(y - 1)

Solve:                                       y  =  -10/17     or     y  =  1

 

Substituting these values back into the equation:  x - 7y  =  2

     x - 7(-10/17)  =  2     --->     x  =  -36/17      --->     Answer:   (-36/17, -10/17)

     x - 7(1)  =  2             --->     x  =  9              --->     Answer:    (9, 1)

 

I believe that when graphed   x²  =  34y² + 7xy  - 16  is a hyperbola  and  x - 7y  =  2  is a straight line, so that their solution, their intersection, will either by no points, one point, or two points; in this case:  two points.

Perhaps someone can post the graph ...