Maximise Z = 5x + 3y subject to x + y ≤ 8, x ≤ 6, x ≥ 0 and y ≥ 0…

Step 1. The corners. The line $x = 6$ meets $x + y = 8$ where $y = 2$. So the corners are $(0, 0)$, $(6, 0)$, $(6, 2)$ and $(0, 8)$.

Step 2. The values. At $(0, 0)$, $Z = 0$. At $(6, 0)$, $Z = 30$. At $(6, 2)$, $Z = 30 + 6 = 36$. At $(0, 8)$, $Z = 24$.

Step 3. Compare. The largest value is $36$, at $(6, 2)$.