help hard algebra

There's a really good inequality for this, Cauchy-Schwarz inequality.

$(a_1^2+a_2^2+\dots+a_n^2)(b_1^2+b_2^2+\dots+b_n^2)\ge(a_1b_1+a_1b_2+\dots+a_nb_n)^2$ , where a and b are a sequence of real numbers.

Applying this, we want to get xy on the right side so we do:

$(x^2+3y^2)(1+\frac{1}{3})=(x+\sqrt{3}(\frac{1}{\sqrt{3}})y)^2$

$18*\frac{4}{3}\ge(x+y)^2$

$x+y\le\sqrt{24}$.

The maximum value is $\sqrt{24}$.

(Yes the equality condition can be satisfied).