# Linear Programming 3: Graphical Solution – with negative coefficients

Welcome! In this tutorial, we will solve this maximization

problem using graphical method. Let’s label the constraints C1 to C4 for

reference purposes. Let’s start by setting up tables to find

the points. The line equation for Constraint 1) is 7X

– Y=3 So when X=0, Y=-3

And when Y=0, X=0.43 These two points are not that useful to us

because the first has a negative which will take us far away from the feasible region

and make the graph very small. The other has a fractional value that cannot

be easily located on the graph. So let’s try finding more points.

To make it easy to find useful points, it is better to rewrite the equation in terms

of one variable. In this case, we can write it in terms of

Y. That is, Y=7X – 3.

We can now use trial and error to find better points.

For example, when X=0.5, Y also equals 0.5. And when X=1, Y equals 4. For Constraint 2) we have the line -3x + 6y

=10 when X=0, Y=1.67.

And when Y=0, X=-3.33. Again, let’s try more points.

Rewriting the equation we have 6Y=10 + 3X so that Y=(10 + 3X)/6.

When X=1, Y=2.17 And when X=2, Y=2.67.

So it is hard to find whole number points for this line. For Constraint 3) we have the line 3X + 4Y

=9 When X=0, Y=2.25.

And when Y=0, X=3. Let’s try one more point:

when X=1, Y=1.5 And finally for Constraint 4) we have the

line 3X + 3Y=3 So when X=0, Y=1

And when Y=0, X=1. And that should do it for that constraint. In drawing the line for the first constraint,

we can just plot the last 2 points (.5,.5) and (1, 4).

And then the constraint line. For the second constraint, we can use the

first point and the last 2. And that’s the line.

For the third constraint, we can use (3, 0) and (1, 1.5); and that’s the line.

And for the 4th constraint, we have (0, 1) and (1, 0); and that’s the line. Now the first 3 constraints are “less than”

constraints and are satisfied in the direction of the origin.

The last constraint a “greater than” constraint satisfied in this direction away from the

origin. The directions of the arrows show that the

feasible region is this area shaded in blue. To determine which of the extreme points of

this feasible region is optimal, let’s use the objective function line method.

The objective function is to maximize -3X + 12Y.

We begin by setting the objective function to any number of our choice.

Normally I will just multiply 3 by 12. But that will give us 36 which will give us a

line too far away from the feasible region. But since 12 is a multiple of 3, it will be

a better choice in this case. So let’s set the objective function equal

to 12. Next we find points to draw the objective

function line. When X=0, Y=1

And when Y=0, X=-4. Because of this -4, let’s just obtain one

more point. When X=2, Y=1.5.

So using these 2 points, the objective function line is this dotted line here. Since this is a maximization problem, we slowly

move the objective function line upwards away from the origin, parallel to itself, to obtain

the optimal solution point. Now moving the objective function line shows

that the optimal solution occurs here at the line intersection of constraints 2 and 3.

Let us now to solve these two lines simultaneously to determine the coordinates at that point.

Here are the equations. Since the coefficients of X in the two equations

are -3 and +3 respectively, we can simply add the 2 equations to eliminate X.

So -3X cancels 3X. 6Y + 4Y gives 10Y,

and 10 + 9 equals 19. On dividing both sides by 10 we have Y=1.9.

Substituting that Y value in C3 we have 3X + 4(1.9)=9.

That is 3X=1.4 And X=0.47.

So the optimal solution is X=0.47 and Y=1.9.

Plugging that point into the objective function we have

-3(0.47) + 12(1.9) which gives 21.4. So the optimal solution is X=0.47 and Y

=1.9 And the corresponding maximum value of the

objective function is 21.4. See you in the next video.

And thanks for watching.

thanx a lot you're the best

Regards

thank you so much your voice and hands

You have a wonderful voice and your videos are so well made and good at explaining the solution! Thanks for the help!!!

3:02: "are less than constraints and are satisfied in the direction of the origin".

Is it just a coincidence that this worked for the

negativecoefficient constraint -3X +6Y <= 10 (purple line)?Because if you try it on this negative constraint: -3x + 2y ≤ -4

You'll realise that even though this is a "less than constraint", it will NOT be satisfied in the direction of the origin (as far as I know, unless I graphed it incorrectly) , like you stated in the video.

Rather, it will be satisfied away from the origin… and the only way to have worked this out is to have input a test coordinate, like you had suggested in the comments earlier. Funnily you didn't do this when graphing the purple line, so was it just a coincidence that you got lucky to know which direction to shade in? Or how did you know exactly without inputting test coordinates?

thanx too much ..I hope that you can also be described the simplex method…in the same charming and understanding way

Thnk u so much for explanation in detail….

best explanation ever!! thank yoU!

thank youuu!!

you helped me in my exams … soo thankful to u

well explained..thank you so much

sir explain simplex method. plz

I couldn't not thank you. so, thank you very very much

Very clearly and thoroughly explained. You make it seem quite easy. Thank you.

Al hail Joshua !!!

i cant thank you enough !!!

Thanks so much for very good explanation. You've helped me to prepare for a midterm. My God will continue to strengthen you and help you in all your endeavours.

not a fucking indian accent finally

hi. sorry but when you say trial & error for 1:07 points, do you mean just randomly think of any number or is there like a range of specific numbers to pick from? can i just randomly use 0.75?

also for the second one, (-3X+6Y=10) why did'nt we use 0.5, why did we go straight to 1? what are we really looking for by getting other values? i thought it was just to offset the negatives. please explain 🙁

sorry last question, for the third constraint, (3X+4Y=9) why didnt we rearrange the equation when we found the third point. i rearranged the equation and when x=1, y=3. which is different than the un-rearranged equation where your y=1.5

Great video

Thank you so much !!!!!! (From South Korea)

i'm korean thankyou for teacher you are helpful to me