Diophantine Equation with 3 Variables Find all solutions to $2x + 3y + 4z = 5$.
I know how to do it with two variables, but I'm confused on how to start this with three variables.
 A: Temporarily, let $x+2z=w$. You know how to find the general solution to $2w+3y=5$. But we review the idea briefly. 
We can see that $w_0=1$, $y_0=1$ works. If $(w,y)$ is any solution, then $2(w-w_0)+3(y-y_0)=0$, and therefore $w=w_0+3t$, $y=y_0-2t$ for some integer $t$. Furthermore, $w=w_0+3t$, $y=y_0-2t$ always is a solution. In our case, the general solution is $w=1+3t$, $y=1-2t$. 
Now write down the general solution of $x+2z=1+3t$. One solution is $x_0=1+3t$, $z_0=0$. So all solutions have shape $x=1+3t+2s$, $z=-s$. 
Now we have all solutions of $2x+3y+4z=5$, in parametric form. Note that there are two free parameters $s$ and $t$.  
A: This might be an ad hoc solution, I don't know if there is a more conceptual way to do it:
First of all, reduce the equation modulo $2$. You get
$$y\equiv 1~(2).$$
So $y$ needs to be odd. If we write
$$y=2u+1,$$
the equation becomes after a small manipulation
$$x+3u+2z=1.$$
So you can express $x$ via $u$ and $z$. The set of solutions should then be:
$$
\{(x,y,z)~|~y\text{ odd, } x=1-2z-3(\frac{y-1}{2})\}
$$
A: 2x + 3y + 4z = 5
2x + 3y + 4z = 5 (Divide by 2, i.e, by the least coefficient among x,y,z)
x + y + y/2 + 2z = 2 + 1/2
y/2 - 1/2 = 2 - x - y - 2z (Separating fractions and integral part)
y/2 - 1/2 = k (constant)
y = 2k + 1
k   y
1   3
2   5
For each value of y you will have to solve the equation for x and y,
after 2 or 3 values you will find a pattern.
