If I remember correctly, to do this problem you need to use substitution. So in the first equation, move the x over to get y=8-x. Then plug that equation into the other one, meaning you replace all y's in the second equation with 8-x. Then you solve for x in the second equation. Once you have the x, you can plug that into the first equation to find the y value.
And if all else fails, graph both equations. The thing to watch out for with linear equations and hyperbolas is that the linear equation might describe one of the asymptotes of the hyperbola, meaning you wouldn't have an intersection.
And you simplified that second equation wrong when you took the square root of both sides. I don't think you can do it like that.
Also, the answers I was looking for were perfect numbers and hexagonal numbers.