Number Theory - Solving an Equation using Modular Arithmetic 
Solve
  $$49x+59y+75z=0 $$
  for $x,y,z\in\mathbb Z$

Please use modular arithmetic. By trial and error I got that the solution is $x=-7, y=2, z=3$. I have solved a similar equation with $2$ variables using modular arithmetic but am unsure how to proceed on one with $3$ variables.
I would like to find the solution stated above, a non trivial solution. 
 A: We have $$49\mid 59y+75z \Longrightarrow 49\mid 10y -23z \Longrightarrow 49\mid 50y -115z \Longrightarrow 49\mid y -17z $$
So we have $y-17z = 49t$ for some $t\in \mathbb{Z}$. Thus $$y = 17z+49t$$ and $$x= -22z-59t$$
where we can choose $z$ arbitrary. 
A: One way you can reduce this a bit, by reducing the coefficients, but giving yourself more work at the end, is to put $a=x+y+z$ so that $$49a+10y+26z=0$$
Now put $b=4a+y+2z$ so that $$9a+10b+6z=0$$
Here you already have $b=3c$ a multiple of $3$ and $a=2d$ is necessarily a multiple of $2$.
$$18d+30c+6z=0$$ or $$3d+5c+z=0$$
Choose $c$ and $d$ as you wish, and $z$ is now determined $z=-3d-5c$.
We have $b=3c$, $a=2d$.
Now $y=b-4a-2z =3c-8d+6d+10c=13c-2d$
And $x=a-y-z=2d-13c+2d+3d+5c=7d-8c$
Putting this together, for any integers $c,d$ there is a solution $$x=7d-8c$$$$ y= 13c-2d$$$$ z=-3d-5c$$
Your solution corresponds to $c=0, d=-1$

For "small" solutions it depends what you mean by small. Any of $x, y, z$ can be made equal to zero, but that gives large numbers elsewhere (excluding the all zero solution).

What you have here is a single linear homogeneous equation in three variables. If $X$ and $Y$ are triples which solve the equation then so does $aX+bY$. Here the solution is given in the form $$(x,y,z)=c(-8,13,-5)+d(7,-2,-3)$$
This has the features we expect - it is linear, and the solution space has dimension $2$ as we would expect having one linear constraint on a three variable linear problem.
I am not sure what advantage you see in using modular arithmetic rather than linear algebra (either ad hoc or general techniques). When you reduce modulo some integer, you still have a linear system to solve, and then you have to lift the solution to the original context.
I have used such observations as $b$ is a multiple of $3$ - which is an implicit use of modulo $3$ for example. Examining the form of the solution, what modulus do you think might help to find that?
