substitute into a set of inequalities

Hello!

Could you please help me with the following problem. Let's say that I have the following equations:

x1+x2<=17

x1+x3<=27

x2+x3<=21

Here we have 3 unknowns. In matrixform: Ax<=b where

A=

1 1 0

1 0 1

0 1 1

b= 

17

27

21

Now I get the new information saying that x1+x2+x3=22 (equation). Then I can isolate x3=22-x1-x2. Then I can substitute this into the previous inequalities, which gives me:

x1+x2<=17

-x2<=5

-x1<=-1

which again can be written in the matrixform Ax<=b where

A=

1 1

0 -1

-1 0

and

b=

17

5

-1

This time there are only 2 unknowns, namely x1 and x2. Once I have them, x3 can be calculated by x3=22-x1-x2. This is just one example. I would like to be able to do this for a general problem. In this example, I would like to give matlab the matrices

1 1 0

1 0 1

0 1 1

and

17

27

21

and the information stating that x1+x2+x3=22.

Then I would like to get the folowing matrices as output:

1 1

0 -1

-1 0

and

17

5

-1

Is it possible to do that in Matlab?

Thank you very much for your time and consideration!

Best regards,

Alex

Answer 1

[Matt J]

One way. Define

    >> T=[1 0;0 1;-1 -1], c=[0;0;22]

    T =

         1     0
         0     1
        -1    -1

    c =

         0
         0
        22

Note that your new equation allows you write [x1;x2;x3] in terms of [x1;x2] in matrix form

 [x1;x2;x3]=T*[x1;x2]+c

substituting into A*x<=b leads to A*(T*x+c)<=b or

    >> Anew=A*T, bnew = b-A*c

    Anew =

         1     1
         0    -1
        -1     0

    bnew =

        17
         5
        -1