Step 1: Checking which constraint is active:
At the optimum for a maximisation LPP, the objective line \(Z=105x+90y\) is pushed as far as possible while staying inside the region, so the optimum sits on the boundary intersection farthest along the direction of \((105,90)\) — expect it near where both constraints are tight.
Step 2: Solving the two tight constraints together:
\(2x+y=80\) and \(x+y=50\): subtracting gives \(x=30\), then \(y=20\), i.e. corner \((30,20)\), consistent with both constraints binding simultaneously.
Step 3: Ruling out the pure-axis corners:
\((40,0)\) gives \(Z=4200\) and \((0,50)\) gives \(Z=4500\) — both less than the value at the binding-constraints corner.
Step 4: Confirming (30,20) is optimal:
\(Z(30,20)=105(30)+90(20)=4950\), the largest of all corner values, and by LPP theory the optimum of a linear objective over a bounded convex region always occurs at a vertex.
Final Answer:
\[ \boxed{\text{Maximum } Z=4950 \text{ at } (30,20)} \]