Diophantine Equation with 3 Variables

Question

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.

Answer 1

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 $t$.

Answer 2

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: $$ \{(x,y,z)~|~y\text{ odd, } x=1-2z-3(\frac{y-1}{2})\} $$

Answer 3

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.