A region is given by 2x + y ≤ 12, x + 2y ≤ 12, x ≥ 0 and y ≥ 0…

Step 1. The meeting point. Adding the two equations gives $3x + 3y = 24$, so $x + y = 8$. Subtracting $x + 2y = 12$ from $2x + y = 12$ gives $x - y = 0$. So $x = y = 4$.

Step 2. The area. Split the region into two triangles from the origin. The triangle with corners $(0, 0)$, $(6, 0)$ and $(4, 4)$ has base $6$ and height $4$, so its area is $\tfrac{1}{2} \times 6 \times 4 = 12$. The triangle with corners $(0, 0)$, $(4, 4)$ and $(0, 6)$ has base $6$ along the $y$ axis and height $4$, so its area is also $12$.

Step 3. Add. $12 + 12 = 24$.