uh hello students once again welcome you to gks maths classes uh today what we are going to do is we are going to solve lpp problem which I've already solved in part one of my video today we are going to solve the same problem but today we are going to use a software called geogebra which is a freely downloadable software and solve this problems okay now for science students who are you have taken only mathematics as a subject I think for them it will be more of an enrichment to cross check whether your answer is right or not but for people who have taken Applied Mathematics you can use this as one of your practical sessions okay so it's up to you so this is actually a very good enrichment to understand how to solve lpp graphically using a software called gebra that is the idea okay so I'm going to solve three problems today using geogebra you can choose any number of problems can you can try to solve any problems using uh gibra for lpp but today what I'm going to do is I'm going to solve three problems one is bounded and unique solution and I going to solve unbounded and infinitely many solution and then unbounded and it has no solution so these three types of questions I'm going to solve okay using geogebra so I will be explaining the steps one by one so you can make a note of it and then you can write it as a procedure and put it in the record book and you can can use it as a practicals okay so let us straight away go inside the first problem okay so I'm going to shift my uh screen from uh this to the system so that you'll be able to see the geogra window and I will be sitting and explaining the concepts I hope it's okay for you okay so let's straight away solve this is a problem which I'm going to solve using geogebra okay so you can make a note of this question now first thing is I need to go to H Mii okay okay let me solve the problems it will take few minutes to load just wa okay now this is the screen this is a gibra window watch it okay now this is gebra I'm using and this is called the input window see right at the bottom this is the input window in the input window I'm going to type the equations okay now what are the equations uh first my first equation is 4x + y 4 x + y is greater than or equal to you see put greater than and then equal to 80 so you get the entire region okay so you are not able to see but there is a shading you see it is there 4x + y greater than or equal to 80 the next one is x + 5 Y is greater than or equal to 115 so that also has been shaded okay so maybe if you go inside and see you'll be able to see that okay maybe I'll zoom out and show you can you see it it's there use you can use a zoom in and zoom out to visualize now the third constraints is what is the third constraint 3 x + 2 y is less than or equal to 15 see and just press enter so that will give you all the three constraints now type the non- negative constraint what is non- negative constraint X is greater than or = to 0 and Y is greater than or equal to 0 x is greater than or equal to 0 and Y is greater than or equal to zero press enter so you have got all the five constraints three constraints and two non- negative constraints now what you do I am going to type the equation of the constraints what are they 4x + y = to 80 so I'll get that line then x + 5 y = to 115 here I'm typing only equations okay 3x + 2 y = 150 then x is = 0 y = 0 I have got all my constraints so now what I'm going to do is I'm going to hide all this INE equation how do I hide go and click on this ring you know this just click on it everything will go away okay so I will hide all the constraints all the inequalities I have I'm going to hide it okay that I have done now I'm going to find the feasible region now how do I find the feasible region now see this inequalities are given a name a b c d e can you see it a b c d and e so in the input box what you do you type a you know it's actually the overlap overlap means and right it's and it's intersection so what you should do is to overlap here the command is double ampon and if you put double ampon and if you put double ampon and it will take it as intersection B again double erson and C double erson and d and double erson and E I hope you understand a and and B and and means intersection a intersection B intersection C intersection D intersection e now you see I got the fusible region you can see it very clearly so sorry this is my fusible region can you see the feasible region this is the feasible region the shading is the feasible region now how do I find the corner points now select the intersection tool can you see here there under this go to the intersection tool click on that now click these two lines first line and second line you get the point of intersection see there's a point of intersection again click on this line click click on the other line it will give you point of intersection that is B now click on this line and click on this line that will give you the third point of intersection is it clear you have got all the and you can see these are the coordinates a b c are the coordinates clear a b c are the coordinates now what I will do I want to know whether it is Max it's actually question is on maximize so I want to type my Z Zed how do I type my Z use P or Q or R whatever alphabet you want capital letter P of X comma y see p of X comma y equal to what is my uh objective function 6X + 3 Y 6x + 3 y press enter now what I want to do I want to find the value of Zed for the corner points so now it's very simple you have to put P of what is the first Corner point a so I got 228 next capital P of B this is the second Corner point for the second Corner point it is 20 150 and P of C this gives me the third Corner point now the question is to maximize so what is the maximum value of Z so clearly 285 is the maximum value can you see it this is the maximum value 285 is the maximum value of Z okay so let me just quickly go through this now this is a gjra window and and this is the input box so first in the input box what I do I type all the inequalities whatever is there in the question inequality including the non- negative constraints you type and after that what you do forget the inequality you type the equality why I'm typing the equality so that I can see this lines otherwise you won't be able to see the line suppose if this not there just see I hide it see you won't see the line you will see only the shading so just to avoid that I'm typing the equation just a second so now you type the equation after that what do you do you hide you hide the inequality after hiding the inequality you type a command a double erson and B Double erson and C double erson and d double ampers n e here double amp n indicates intersection so a intersection B intersection C intersection D intersection e which is the region common to all the four in all the five inequality that gives me this triangle which is a bounded region that is why I say the feasible region is bounded how do you find Corner point you can go to the intersection tool here click on the two lines click on these two lines click on these two lines you get all the points of intersection once you get point of intersection you have to Define find your Z function you use cap I have used capital p x comma y you can use capital Q capital A anything you want is 6X + 3 y now what I want to do I want to find what was my Zed value for this corner points so you just type P of capital A P of capital B P of capital c that will give you these values in this which is the maximum the maximum value is 285 so the question the answer is the maximum value of Z is 285 for x is equal to for X is = to 40 and Y = 15 so it's a very very very simple question is that clear now I'm going to go to the next problem let me show you the problem first okay so we have solv this bounded and unique solution what's the next problem the next problem I'm going to do is unbounded unbounded and infinitely many solution so you know last class I also spoke about what you should do for unbounded region where you have to draw an open off plane so I'm going to show you all that how to do the open off plane you are able to see visually the open of plane and also the unbounded region and see whether there is points in common so we'll check that okay so let me go to the problem let me first go to the HDMI okay just give me few minutes settle down okay now I'll open a new window so what are the constraints first thing is as now I got a new window okay in the input box in the input box I'm going to type I'm going to type the constraint what is the first constraint 3x + 4 Y is is greater than or equal to 8 this is my first constraint 3x + 4 Y is greater than or equal 8 Now what is my second constraint my second constraint is 5X plus 4 y sorry 2 Y 2 Y is greater than or equal to 11 okay now this is my second constraint what is my third constraint non- negative condition X greater than or equal to0 and y greater than or equal to Z so just like what we did in the previous problem I am able to write all the constraints with the inequality now what is the next step type the equality 3x + 4 y = 8 this is the equation now what you should do 5x + 2 y = 11 what's the third one x = 0 and Y = 0 y = to Z so I have typed all the equations now what is the next step as I told you before hide all this first let me hide it I have hidden all the inequalities not the equality but the inequalities then I want to find the feasible region so what do you do a double erson and B Double ampers and C double erson and D so clearly you can see the feasible region is unbounded can you see it the feasible region is unbounded previous problem the feasible region was bounded here the fusible region is unbounded okay now I need to find the corner points now how do I find the corner points so I want to find the corner points so for that go to intersection tool see from here go to intersection tool click on the first line and then click on the Y AIS that will give you this point you have got the point a a is 0a 5.5 now click on this line and this line that will give you the intersection of these two lines that is point B can you see it that is point B that is 2 comma 0.5 then click this line and the xaxis that will give you the third Point C so a b c are the corner points are the corner points now what I'm going to do I'm going to find my Z value so first I have to type P capital P of X comma y equal to what is my Z function 60x plus 80 y 60 x + 80 y 60 x +8 8 y right now capital P of a what is capital P of a 440 you can cross check it by doing manually also now capital P of what capital B vertex B what is that 160 and capital P of C what is that again it is 160 you see it has two same values remember when we are doing when it has two same values it should have infinite many solution you know the question is to minimize here the question is to minimize so the minimum value is 160 but 160 is happening at b as well as C okay but since the feasible region is unbounded you need to draw the open half plane now I'm going to type the open half plane now what is this open of plane the open of plane is nothing but 60x so now I'm typing open of plane 60x + 80 y remember it's a minimization problem so it should be strictly less than can you see it is strictly less than what is the minimum value here the minimum value is 160 correct press enter can you see there a dotted line can you see the dotted line maybe I'll just zoom in and show you see it's a dotted line can you see it okay so this is the open half plane now clearly you can see open of plane and the the blue region and the pink region have no point in common have no point in common therefore 160 is the minimum value okay so let us see one more thing what I will do is I will try to draw the open of plane for you I will hide this for some time see I will hide I will hide this okay I will say open of plane how do I get X greater than zero that is C double erson and D that is X and Y and double erson and J J is my J is my open of plane and I press enter now you can see the open of plane and the feasible region that is unbounded have no point in common therefore 160 is the minimum value for what value of X and Y for All Points X and Y in between BC I'll just show you that for All Points between X and Y so what I will do I'll choose a point point on the object can you see point on the object and I will click anywhere in between BC and that point is D now I will try to find P of D can you see p of D see it is also 160 now what I will do I will try to move this D you see that value will never change look at this 6 160 it will never change it will never change can you see it I moving it it never changes B keeps changing but my Z never changes so when you have two same minimum value okay and if the open of plane have no point in common with the visible region then 160 is the minimum value for all points in the line segment joining BC I have demonstrated that okay so now let me once again explain this problem to you what did I do here first I typed the inequality okay as usual then I type the equation of that inequality with the equal to sign then I hide all the inequalities I hide by just clicking that dark dot it will hide by itself and then what I do I want to find the feasible region so I want to know what is common between a a b c and d so what I do a double ampon b double ampon and C double ampon and D again I'm mentioning to you that double ampon and stands for intersection so when you do that you get the pink region can you see this is the pink region which is unbounded okay then what you do you want to find point of intersection so you can go here and find point of intersection you see this is a point of intersection two okay once you do select that you all that you need to do is see where it intersects okay let me show you that so I find all the points of intersection I want to know where it intersects the x axis Y axis so this line and Y AIS then select these two lines you get this point select this line and this line get all the points a b and c so this are the corner points then what I do I type P of X Comm y this is my Z function which is 60x + 80 y then I find P of a p of B P of C which happens to be 440 160 and 160 this is a question to minimize since the question is to minimize my 160 will be the minimum value but since the phical region is unbounded I need to draw the open off plane now how do I draw the open off plane Z function that is 60x + 80 y strictly less than 60x + 80 y strictly less than 160 when I do that I'll just show it to you when I do that you see I get can you see the dotted line it's blue dotted line okay that means the points on this B this B do not belong to J even though you can see B on the line this B belongs to the unbounded region B belong do not belong to J because it is strictly less than when you say less than line don't belong to the inequality so I'll remove it okay I don't want it now what do you do I want to know what is common between x-axis y axis and this inequality so C double ampon and d double ampon and J gives me the open of this is the open of plane now you can see this brown region is open of plane pink region is the unbounded they do not overlap they don't have any point in common since they don't have any point in common 160 is the minimum value but there is no 160 but there are 260s at point B and C that means it is the entire line joining it is all the points joining BC let me just show you that part also once again so I'm zooming in now have an eye on this 160 okay so when I change B when I change B it will never change so BC from the point B to C okay from point B to C all the points are taken as solution so this is unbounded and it has infinitely manyu solution what is infinitely menu solution All Points joining B and C is it clear okay let's go to the last problem one more question I'll do I will show you the problem first this is the problem we did unbounded and infinitely many solution now the third one is unbounded and it has no solution unbounded and no solution so and this is the question just have a look at the question okay now we are going to solve the problem using geogebra let's go to the geogebra HDM now what is the last problem okay let's see first let me type the inequalities in the in in the input box first I type all the constraint what are the first constraint x + y greater than or equal to 5 x + y greater than or equal to 5 what happened have come just a second I'm I'm typing it okay x + y greater than or equal to 5 okay this a problem now what is the second constraint my second constraint is x + 2 y x + 2 y is greater than or equal to 6 x + 2 y is greater than or equal to 6 I have three third constraint X is greater than or equal to 3 so there are totally three constraints now non- negative constraints X greater than or equal to z and y greater than or equal to Z so all the three constraints are written okay so totally three constraints and two non- negative constraints now I type the equation x x + y = 5 then x + 2 y = 6 then x = 3 x = 0 y = 0 all the equations are typed what is the next step hide hide the INE equation so I'm hiding it now I need to find the overlap so how do you find overlap fusible region a double ampon and B Double erson and C double ampon and d double erson and E so I got everything so it's again unbounded can you see it the entire blue region is completely unbounded so now I need to find the the point of intersection I need to find the corner points so it's a intersection of this line and X is equal to 3 that gives you point a that is 3A 2 then it is intersection of this line and the second line what is that that is B now it is intersection of this line with the x-axis so there are totally Three Corner points correct a b and c I think you can visualize it there are three corner points a b and c now I want to type my Z function what is my Z function P of X Comm y = - x + 2 y this is my Z - x + 2 y now I want to find P of a what is p of a 1 then P of B what is p of B it's -2 then P of C what is that it is - 6 now here the question is to maximize question is to maximize so what is the maximum value of Z the maximum value of Zed is one so I need to type the open half plane why do I have to type the open half plane because the feasible region is unbounded how do I type the open off plane take the Z function - x + 2 y - x + 2 y is greater than y greater than the question is to maximize equal to 1 why do I type greater than one because one is the maximum value when I type enter you see you got a dotted line can you see dotted line okay open half plane okay now what I will do I want to see what is my open of plan is so it is the intersection of what it is intersection of C that is X is see when you say x is greater than zero and then X is greater than okay I will take X is greater than zero okay that is d double erson and E and double erson and K that is the open of plane all right now I'll now I'll hide the C now you can see clearly I just see see here clearly just a second uh now you can see clearly the blue region this is the entire blue region and the open of plane have points in common if the open of plane have point in common with the the feasible region then one is not the maximum value therefore this problem cannot be maximized so it has no solution for this particular question the feasible region is unbounded and it has no solution okay again I just go through this once again just watch it first I type all the inequalities then I Tye the corresponding equalities after doing that next step is I hide all the inequality then I want to find the fible region so I go for a double amp and B Double ampon and C double ampon and d double ampon and E that will give you this entire blue region this is my entire full blue I'm sorry okay I get the entire blue region so this is the blue region can you see it the entire blue region completely get that's the fible region then you go to the intersection tool okay you go to the intersection tool and find the intersection intersection of x isal 3 and this line intersection of these two lines and intersection of this line on the x-axis that I think you can get it once you got the intersection you define your Z function which is - x + 2 y then you put P of a p of B P of c and you see p of a is 1 which is the maximum value the question is to maximize now now I have to draw the open of plane what is open of plane - x + 2 y strictly greater than 1 can you see the dotted line strictly greater than one one I can show you once again see this the rted line the entire that so now what I want to do I want to find the open of plane so how do I find open of plane I'll take X greater than 0 y greater than 0 and the open off plane and I got this pink region can you see this pink region now you can see clearly the pink region overlaps with the blue region since the pink region and blue region have points in common H equal to 1 is not the minimum value because you know very well when we did problems last class I have mentioned very clearly that if you have an unbounded region the open of plane should not have any point in common with the fible region if it has point in common with the fible region it has no maximum if it a maximization problem and it has no no minimum if it is a minimization problem so in this case even though the feasible region is unbounded and since the open of plane have points in common with the fible region the question is to maximize so one is not the maximum value okay so with that I will stop I think we have done three different types of questions where three different difficult situations where you are able to visualize these problems okay so for science students again those who have taken mathematics only for for them you know this will be more of an enrichment to cross check whether you are getting the whe whether you're proceeding in the right direction or Not by using these tools but for Applied math students what I advise is you can always use this as one of your practicals okay you can put it in your record book you can take any one problem or two problems or all the three problems or you can create your own problems and try to solve it actually it's very interesting it gives you a better understanding of what you're doing okay I hope this video is very useful to you we'll continue you in our next video kindly subscribe to my channel and also click the Bell icon to get notifications have a wonderful day see you again in the next class bye-bye