mixed system of equalities and inequalities, for example: x + 2 y + 3 z = 10 2 x + 4 y + 10 z = 20 4 x + y + z &lt; 10where unknown variables x , y , z are all real-valued.How can get a solution (or the range of all feasible solutions) for this system?

Question

mixed system of equalities and inequalities, for example:  x  +  2  y  +  3  z  =  10    2  x  +  4  y  +  10  z  =  20    4  x  +  y  +  z  &amp;lt;  10where unknown variables   x  ,  y  ,  z are all real-valued.How can get a solution (or the range of all feasible solutions) for this system?

Jordin Church · Accepted Answer

Multiply the first equation with 2, the eq.1 and eq. 2 read:  2  x  +  4  y  +  6  z  =  20    2  x  +  4  y  +  10  z  =  20  Subtraction gives:   4  z  =  0 hence   z  =  0.From the first eq. we now derive   x  =  10  −  2  y  . Thus the Third eq. becomes:   4  (  10  −  2  y  )  +  y  &amp;lt;  1  = or   y  &amp;gt;      30    7    .Consequence: the set of solutions is given by:  {  (  10  −  2  y  ,  y  ,  0  )  :  y  &amp;gt;      30    7    }  .

Lucian Maddox · Accepted Answer

I see, thank you. My problem is my system is usually composed of M equations and N inequalities (and M+N variables) where both M and N can be very large, making an analytical solution potentially tedious