Solve the following system of equations (1)

Question

Solve the following system of equations: $$\large \left \{ \begin{aligned} \sqrt{x + 2}(x - y + 3) &= \sqrt{y}\\ x^2 + (x + 3)(2x - y + 5) &= x + 16 \end{aligned} \right.$$

That is definitely not easy to solve for me. I try to solve the question by letting $x + 2 = a$ and $x - y + 3 = b$ but it didn't work.

Answer 1

Let $a=\sqrt{x+2}$ and $b=\sqrt{y}$, then first equation is $$a^3+a-ab^2=b$$ so $$ a(a-b)(a+b)+a-b=0$$ and thus $$(a-b)(a^2+ab+1)=0$$

Case 1: $a=b$ so $y=x+2$:...

Case 2: $a^2+ab =-1$ is impossible since $a,b\geq 0$.

Answer 2

Hint: Squaring the first equation and factorizing we get $$(x-y+2)(x^2-xy+6x-2y+9)=0$$ so we get $$x-y+2=0$$ or $$x^2-xy+6x-2y+9=0$$ Can you proceed now?