Absolute Confusion

If we add the first two equations, we get

$|x| + |y| + 2x =1$ .

Since we are looking for integer solutions, this means that we may conclude that $x<0$ . Further manipulation of the equations does not allow us to conclude directly whether $x$ or $y$ must be positive or negative. The system of equations depends on whether the values of $x$ and $y$ are positive or negative.

If there is a solution, it must be among the following four cases:

a) $x<0, y>0, z>0$

b) $x<0, y>0, z<0$

c) $x<0, y<0, z>0$

d) $x<0, y<0, z<0$ .

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Rewriting the equations for each case gives:

a) $y-z=19; x+z=-18; x-y=19$ $\Rightarrow (x,y,z)=(10,-9,-28)$

b) $y-z=19; x+z=-18; x-y-2z=19$ $\Rightarrow (x,y,z)=(-4,5,-14)$

c) $y-z=19; x-2y+z=-18; x-y=19$ $\Rightarrow$ not solvable

d) $y-z=19; x-2y+z=-18; x-y-2z=19$ $\Rightarrow (x,y,z)= (11,10,-9)$ .

We see that only case b) satisfies the conditions: $x=-4, y=5,z=-14$ .

Thus, $xz-y= (-4)(-14)-5= \boxed{51}$ .

The full solution is pretty lengthy, so I'll only put up a partial solution. The way to do this problem is to assume signs of each of the x, y, and z variables. That means there are 8 possible assumptions:

$\begin{matrix} \underline { x } & \underline { y } & \underline { z } \\ + & + & + \\ + & + & - \end{matrix}\\ \begin{matrix} + & - & + \\ - & + & + \\ + & - & - \end{matrix}\\ \begin{matrix} - & - & + \\ - & + & - \\ - & - & - \end{matrix}$

An important thing to know is the $sgn$ function: $\quad \left| x \right| \quad =\quad x*sgn\left( x \right)$

So if we assume x to be positive, then: $\quad \left| x \right| \quad =\quad x,\quad if\quad x\quad >\quad 0$

If we assume x to be negative, then: $\quad \left| x \right| \quad =\quad -x,\quad if\quad x\quad <\quad 0$

After eliminating some assumptions either by contradiction or by checking by plugging in the value to the system, only - + - for x, y, and z respectively, works, and we arrive at

$x\, =\, -4\\ y\, =\, 5\\ z\, =\, -14\\ xz\, -\, y\, =\, -4(-14)\, -\, 5\, =\, 56\, -\, 5\, =\, \boxed { 51 }$

Let me know if I need to get into further detail.