how to solve a Maple problem with three variables?

Question

3x−4y=9 becomes {x=3+4_Z1,y=3_Z1}

I don't understand where the y=3_Z1 comes from.

How to write this for 3x-4y-5z=9?

Could I write this:

3x-4_Z1-5_Z2, then what?

Answer 1

If you let $y=w_1$ and $z=w_2$, then you have

$$3x+4w_1+5w_2=9$$

Solving for $x$, this gives

$$x=3-\frac{4}{3}w_1-\frac{5}{3}w_2$$

We don't actually know what $w_1$ or $w_2$ are as there are an infinite amount of ordered triples that satisfy the equation. But letting $w_1$ and $w_2$ be any number allows us to solve for $x$ in terms of $w_1$ and $w_2$.

By the way, the reason that Maple uses $y=3z_1$ is that the lowest common multiple of the coefficients $3$ and $4$ is $12$. When you let $y=3z_1$, and solve for $x$, you can see that $x=3-4z_1$, and the solutions involve integer coefficients and not fractions like my solution above.

In lieu of this, you could have let $z=12w_2$ and $y=15w_1$ since the LCM of $3,4$, and $5$ is $60$. Note that the equation then becomes

$$3x+4(15w_1)+5(12w_2)=9 \Rightarrow 3x+60w_1+60w_2=9$$

Then solving for $x$ provides the solution set as

$$\{3-20w_1-20w_2, 15w_1, 12w_2\}$$