# Find the unique solution

#### anemone

##### MHB POTW Director
Staff member
Consider the 3 simultaneous equations

$$\displaystyle |x|-y-z=5$$

$$\displaystyle x-|y|+z=-9$$

$$\displaystyle x-y+|z|=-1$$

Find the unique solution of the simultaneous equation in which $x, y, z$ are all integers.

#### Jester

##### Well-known member
MHB Math Helper
My solution
Maybe not the most elegant but effective. Due to the presense of the absolutes we can consider 8 cases depending on whether $x, y$ and $z \ge 0$ or $< 0$. However we can rule out 6 of these immediately due to inconsistencies in the system. For example, if $x,y,z \ge 0$ the the system is

\begin{eqnarray}
x - y - 5 &=& 5\\
x-y+z &=& -9\\
x - y + z &=& -1
\end{eqnarray}

and clearly the last 2 equations are inconsistent. The only cases I found that were consistent where

\begin{eqnarray}
(i) & \;\;&x \ge 0,\;\;& y < 0,\;\; &z \ge 0\\
(ii)& \;\;&x <0.\;\; &y \ge 0,\;\;& z < 0
\end{eqnarray}

In each case, the systems were easily solved giving solutions $(-2.-4,-3)$ and $(-3,2,-4)$.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
My solution
But you have 2 solutions, when surely there should only be one?! Can you scratch the first one please?
(I selected the same method.)

Last edited:

#### Jester

##### Well-known member
MHB Math Helper
My solution
Maybe not the most elegant but effective. Due to the presense of the absolutes we can consider 8 cases depending on whether $x, y$ and $z \ge 0$ or $< 0$. However we can rule out 6 of these immediately due to inconsistencies in the system. For example, if $x,y,z \ge 0$ the the system is

\begin{eqnarray}
x - y - z &=& 5\\
x-y+z &=& -9\\
x - y + z &=& -1
\end{eqnarray}

and clearly the last 2 equations are inconsistent. The only cases I found that were consistent where

\begin{eqnarray}
(i) & \;\;&x \ge 0,\;\;& y < 0,\;\; &z \ge 0\\
(ii)& \;\;&x <0.\;\; &y \ge 0,\;\;& z < 0
\end{eqnarray}

In each case, the systems were easily solved giving solutions $(-2.-4,-3)$ and $(-3,2,-4)$.
As I like Serena pointed out, there is a unique solution. Clearly my first solution violates the condition $x \ge 0$ and $z \ge 0$.

#### anemone

##### MHB POTW Director
Staff member
Hi Jester,

Thanks for participating and I think your method is quite elegant and I used the same concept to solve it too.

Hi I like Serena,

Thanks for everything!

But I have thought of another method to tackle the problem and the idea only hit me moment ago.

We're given to solve for the unique solution of the simultaneous equation below in which $x, y, z$ are all integers.

$$\displaystyle |x|-y-z=5$$---(1)

$$\displaystyle x-|y|+z=-9$$---(2)

$$\displaystyle x-y+|z|=-1$$---(3)

 Adding the equations (1) and (2) gives Adding the equations (1) and (3) gives Adding the equations (2) and (3) gives $$\displaystyle |x|+x=|y|+y-4$$ $$\displaystyle |x|-x=|z|+z+6$$ $$\displaystyle |y|-y=z-|z|+8$$ Let $x<0$. We then have $$\displaystyle 0=|y|+y-4$$ This implies $y=2$. When $y=2$, we have $$\displaystyle 0=z-|z|+8$$ This gives $z=-4$. When $x<0$, $y=2$, $z=-4$, $$\displaystyle |x|-x=|z|+z+6$$ becomes $$\displaystyle -2x=6$$ $x=-3$ Now, we have to consider the case when $x>0$. When $x>0$, we have $$\displaystyle 0=|z|+z+6$$ And obviously the equation above has no solution and we're done! Yeah! 