I have a mixed system of equalities and inequalities, for example:
$x + 2y +3z = 10\\ 2x + 4y + 10z = 20\\ 4x + y + z < 10$
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? I understand this problem can be converted into a linear constrained optimisation problem, but I do not understand how.
Please help
Multiply the first equation with $2$, the eq.1 and eq. 2 read:
$2x + 4y +6z = 20\\ 2x + 4y + 10z = 20\\$
Subtraction gives: $4z=0$ hence $z=0.$
From the first eq. we now derive $x=10-2y.$ Thus the Third eq. becomes: $4(10-2y)+y<1=$ or $y > \frac{30}{7}.$
Consequence: the set of solutions is given by:
$$ \{(10-2y, y,0): y > \frac{30}{7}\}.$$