Let x be the number of nuts and y be the number of bolts that we can make. Our problem is to maximize x and y if the factory worked at capacity of 12h / day. We also have to find out the maximum profit at this capacity.
Clearly, x, y ≥ 0. Let us construct the following table from the given data

Nuts (x) 
Bolts (y) 
# hrs avail 
Machine A (h) 
1 
3 
12 
Machine B (h) 
3 
1 
12 
Profits (Rs.) 
17.5 
7 

Since we have only 12h of machine time, we have the following constraints:
x + 3y $\leq$ 12
3x + y $\leq$ 12
The profits on the nuts is Rs. 17.5 and on the bolts is Rs. 7. We need to maximize the profits, i.e. maximize 17.5x + 7y, given the above constraints.
$\textbf{Plotting the constraints}$:
Plot the straight lines x + 3y = 12 and 3x + y = 12.
First draw the graph of the line x + 3y = 12
If x = 0, y = 4 and if y = 0, x = 12. So, this is a straight line between (0,4) and (12,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is $\leq$ 0. So the area associated with this inequality is bounded towards the origin
Similarly, draw the graph of the line 3x + y = 12.
If x = 0, y = 12 and if y =0, x = 4. So, this is a straight line between (0,12) and (4,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is $\leq$ 0. So the area associated with this inequality is bounded towards the origin.
$\textbf{Finding the feasible region}$:
We can see that the feasible region is bounded and in the first quadrant.
On solving the equations x+3y = 12 and 3x + y = 12, we get,
x + 3 (123x) = 12 $\to$x + 36 – 9x = 12 $\to$8x = 24 $\to$x = 3.
If x = 3 then y = 12 – 9 = 3.
$\Rightarrow x = 3, y = 3 $
Therefore the feasible region has the corner points (0,0), (0,4), (3,3), (4,0) as shown in the figure.
$\textbf{Solving the objective function using the corner point method}$
The values of Z at the corner points are calculated as follows:
Corner Point 
Z = 17.5x + 7y 
(0,0) 
0 
(0,4) 
28 
(3,3) 
73.5 (Max Value) 
(4,0) 
70 
$\textbf{ The maximum profit we can make is Rs. 73.5, which involves making 3 packages of nuts and bolts each. }$