Simple Question on Maximizing Point on linear two-variable function

Question

Fix $N \in \mathbb{N}$. How would I maximize $f(x, y) := \frac{1}{4}x + \frac{1}{3}y$ given that $x + y = N$ and $x, y \geq 0$ ? I've looked at this, but the second link in the answer doesn't work! The KKT conditions that the answerer references are for non-linear systems, so I'm not sure how that even helps the OP of that post. Anyway, all that I've tried so far, sadly, is substitution and GeoGebra... Thanks.

Answer 1

You can exploit a simple property of linear problems. The solution is one of the intersections of the constraints.

These intersections are $(0,0)$, $(0,N)$ and $(N,0)$.

The one that maximizes $f$ and satisfies $x+y=N$ is $(0,N)$. Thus the solution is $(0,N)$.