glitinosim3

2022-06-30

mixed system of equalities and inequalities, for example:
$x+2y+3z=10\phantom{\rule{0ex}{0ex}}2x+4y+10z=20\phantom{\rule{0ex}{0ex}}4x+y+z<10$
where 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

Expert

Multiply the first equation with 2, the eq.1 and eq. 2 read:
$2x+4y+6z=20\phantom{\rule{0ex}{0ex}}2x+4y+10z=20\phantom{\rule{0ex}{0ex}}$
Subtraction gives: $4z=0$ hence $z=0.$
From the first eq. we now derive $x=10-2y.$ Thus the Third eq. becomes: $4\left(10-2y\right)+y<1=$ or $y>\frac{30}{7}.$
Consequence: the set of solutions is given by:
$\left\{\left(10-2y,y,0\right):y>\frac{30}{7}\right\}.$

Expert