Question

Maximize p = x + y
subject to x + 3y <=4
3x + y<= 4
x >=0, y >=0

Answer 1

You would want to plot the 4 lines to make a convex polygon, inside of which is the feasible region, i.e. any point inside the polygon will satisfy the constraints.

Now plot the line Z=x+y, which is a line at 45 degrees sloping downwards to the right. Draw lines parallel to this line until the line touches on the last corner (top right, approximately) of the polygon. The x-y values of this corner is the answer you require.

You can do all this algebraically, but it is a little more abstract. You are better off starting it graphically and then do the algebraic part afterwards to confirm or get better accuracy.