You can't. Well, you can, in a sense. But you won't have some nice, clean "solution" that can be written on paper.
You have 5 unknowns, in the 5 variables {x,y,a1,a2,a3}. You have 3 equalities (two nonlinear, one linear), and a few inequalities that really do not bound things very much anyway. What matters is that if any solutions do exist, there will generally be infinitely many such solutions, that will live on some curved surface in that 5-dimensional space of your unknowns.
Since your equations are of a high enough total degree that almost certainly no analytical solution exists, the only description of that solution locus will be the set of equations you already have.
You can decide to solve it parametrically, choosing perhaps values of a2 and a3 such that both are positive. That will allow you to choose a1 to satisfy the linear constraint exactly. Then, given numeric values for a1,a2,a3, solve for x and y using a tool like fsolve or vpasolve. Note that zero, one, or more solutions may well exist, although fsolve will generate only one solution for any starting values (x0,y0). (Generally it would be a finite number of solutions.) Since you are restricting the solutions to real variables, it is even more difficult to predict how many solutions exist.
