A number theory problem by Kian Chua

Rewrite the equations as:

x + y -3 = 3u

x - 4y + z = 4u

x - 5y - 5z = -u

Then solve for x and x = 83u/17.. Therefore the smallest integer for u is 17 and that of x = 83.

Dividing the three equations by u ,we get

$\frac{x}{u}$ + $\frac{y}{u}$ = $\frac{3z+3u}{u}$

or, $\frac{x}{u}$ + $\frac{y}{u}$ = $\frac{3z}{u}$ +3

$\frac{x}{u}$ + $\frac{y}{u}$ - $\frac{3z}{u}$ =3

Similarly from equations 2 and 3 we get,

$\frac{x}{u}$ + $\frac{z}{u}$ - $\frac{4y}{u}$ = $4$

and

$\frac{x}{u}$ - $\frac{5y}{u}$ - $\frac{5z}{u}$ = $-1$

respectively.

Now replacing $\frac{x}{u}$ , $\frac{y}{u}$ and $\frac{z}{u}$ by a,b and c in the three equations we get

$a+b-3c=3$

$a-4b+c=4$

$a-5b-5c=-1$

From the three equations,we find the value a to be $\frac{83}{17}$

So the value of $\frac{x}{u}$ is $\frac{83}{17}$ And hence the value of $x$ is $\boxed{83}$ .