4x^2 + 9y^2 = 72,   (1)
x - y^2 = -1.       (2)

To solve it algebraically, express y^2 = x+1 from the equation (2) and substitute it into equation (1). You will get

4x^2 + 9(x+1) = 72.

In this way you got a single equation for only one unknown x.

Simplify and solve it:

4x^2 + 9x - 63 = 0,

 =  = .

 =  = 3,   =  = -5.25.

The negative root doesn't suit the equation y^2 = x+1, since the right site would be negative.

So, the only solution for x is x=3.

Then you get two solutions for y: y^2 = x+1 = 3+1 = 4 and   = 2,   = -2.

Answer. The system has two solutions (x,y) = (3,2)  and  (x,y) = (3,-2).