Problem

A linear programming problem in $x$, $y$ and $z$ is described as follows.

Maximise $P = -x + y$

subject to

$x + 2y + z \leq 15$ $3x - 4y + 2z \geq 1$ $2x + y + z = 14$ $x \geq 0, \quad y \geq 0, \quad z \geq 0$

(a) (i) Eliminate $z$ from the first two inequality constraints, simplifying your answers.

(ii) Hence state the maximum possible value of $P$.

Given that $P$ takes the maximum possible value found in (a)(ii),

(b) (i) determine the maximum possible value of $x$.

(ii) Hence find a solution to the linear programming problem.