using the graphical method solve the following linear programing problem maxmise objective function p=3x+y.subject to following constrains 6x+4y<48:3x+6y<42 x>0;y>0

Graphaical method, a piece of cake.

Y upward axis, x horizontal.

plot the line 6x+4y=48
y=-3/2 x+12 Your area of interest will be blow this line.
Plot the line 3x+6y<42 or
y=-1/2 x +7
your area of interest will be below this line.
Mark the lines x=0, y=0, that gives you four lines, which encloses an area of possible solution.
Now a neat theorem of linear programming is that your objective function will be optimized at one of the corners. So test each corner for the value of P. Wherever max, that is the solution.
Neat and tidy, like Red Velvet Cake.

To solve this linear programming problem using the graphical method, we need to follow these steps:

Step 1: Graph the inequality constraints.
Step 2: Identify the feasible region.
Step 3: Plot the objective function.
Step 4: Determine the optimal solution.

Let's go through each step in detail:

Step 1: Graph the inequality constraints.
Start by graphing each constraint as an inequality on a coordinate plane. To graph an inequality, rewrite it as an equation and then draw the corresponding boundary line.

Constraint 1: 6x + 4y < 48
To graph this constraint, rewrite the inequality as an equation:
6x + 4y = 48
Now solve for y to obtain the boundary line equation:
y = (48 - 6x) / 4
Plot this line using two points or by finding the intercepts.

Constraint 2: 3x + 6y < 42
Rewrite the inequality as an equation:
3x + 6y = 42
Solve for y to obtain the boundary line equation:
y = (42 - 3x) / 6
Plot this line.

Step 2: Identify the feasible region.
The feasible region is the area in the graph where all inequality constraints are satisfied simultaneously. In this case, it will be the region below both constraint lines.

Shade the area below both lines to identify the feasible region.

Step 3: Plot the objective function.
Now, plot the objective function p = 3x + y on the same coordinate plane. To do this, choose some values for x and y and calculate the corresponding value of p. Then plot these points and connect them to form a line.

Step 4: Determine the optimal solution.
To find the optimal solution (maximum value of p), we need to identify the point where the objective function line intersects the boundary of the feasible region. This point will give us the values of x and y that maximize p.

Locate the highest point where the objective function line intersects the feasible region. The coordinates (x, y) of this point will give you the optimal solution.

That's it! You have now solved the linear programming problem using the graphical method and determined the optimal solution.