This is not a system of quadratic equations, but rather a system of two linear equations with two variables, x and y. We can solve it using the method of substitution or elimination:

Method of substitution: From the first equation, we have x = y + 9. Substituting this into the second equation, we get (y + 9)y = -10, which simplifies to y^2 + 9y + 10 = 0. This is a quadratic equation that can be factored as (y + 1)(y + 10) = 0, giving us y = -1 or y = -10. Substituting these values back into x = y + 9, we get x = 8 or x = -1, respectively. Therefore, the solution to the system of equations is (x, y) = (8, -1) or (-1, -10).

Method of elimination: Multiplying the first equation by y and the second equation by (y+1), we get yx - y^2 = 9y and yx + xy = -10y - 10. Adding these equations together, we get 2xy - y^2 = -10y - 10. Rearranging, we get y^2 - 2xy - 10y - 10 = 0. This is a quadratic equation that can be factored as (y - 1)(y + 10) = 0, giving us y = -10 or y = 1. Substituting these values back into the first equation, we get x = 8 or x = -1, respectively. Therefore, the solution to the system of equations is (x, y) = (8, -10) or (-1, 1).