A potter makes x bowls and y vases in a week, both in whole numbers…

Step 1. The corner. Subtracting the two equations gives $x - y = 0$, so $x = y$. Then $11x = 42$ and $x = y = \dfrac{42}{11} = 3.818$. The profit is $70 \times 3.818 = 267.27$, which is $267$ rupees to the nearest rupee.

Step 2. Rounding fails. The plan $(4, 4)$ needs $7(4) + 4(4) = 44$ hours of throwing, which is more than $42$. So it is not allowed. Rounding down gives $(3, 3)$ with profit $210$ rupees, which is much less.

Step 3. Test whole-number plans. $(5, 1)$ gives $230$. $(4, 3)$ gives $250$. $(3, 4)$ gives $240$. $(1, 7)$ gives $250$. $(2, 6)$ uses $7(2) + 4(6) = 38 \le 42$ hours of throwing and $6(2) + 5(6) = 42 \le 42$ hours of glazing, and gives $80 + 180 = 260$. No whole-number plan gives more.

Step 4. Say it. The best whole-number plan is $2$ bowls and $6$ vases, with a profit of $260$ rupees. It is $7$ rupees below the corner value, and not near the rounded corner.