Why is this particular substitution made? ($y=tx$)

Question

A problem in Complex Numbers (Andreescu, Andrica) involving a Complex Equation & a "Substitution" technique :

Let us solve the equation $z^3 = 18 + 26i$, where $z = x + yi$ and $x$, $y$ are integers.

Solution given :

We can write $(x + yi)^3 = (x + yi)^2 \times (x + yi)$ = $(x^2 − y^2 + 2xyi)(x + yi) = (x^3 − 3xy^2) + (3x^2y − y^3)i = 18 + 26i$

Using the definition of equality of complex numbers, we obtain
$x^3 − 3xy^2 = 18$,
$3x^2y − y^3 = 26$
Setting $\color{maroon}{y = tx}$ in the equality $18(3x^2y − y^3) = 26(x^3 − 3xy^2)$, let us observe that $x \ne 0$ and $y\ne 0$ implies
$18(3t − t^3) = 26(1 − 3t^2)$, which is equivalent to
$(3t − 1)(3t^2 − 12t − 13) = 0$.
The only rational solution of this equation is $t=\frac{1}{3}$ ...

I'm not able to understand the rationale behind substituting $y=tx$.
I tried substituting $y=mx+c$ after differentiating both relations, but that leads nowhere ($1=1$, haha).

On thinking about it a bit more, I think it's because the relations are just changes of variables.
It is that $x^3 - 3xy^2$ would become $y^3-3x^2y$ when $x$ and $y$ are interchanged?
Is there some property accounting for this? If there is, could you please explain the rationale?

Answer 1

Every Complex Number $z=x+iy$ can be put on the Complex Plane with Cartesian Coordinates $z=(x,y)$ & can written in Polar Coordinates like $z=(r,\theta)$.
Here $\tan(\theta)=y/x$ , which "Directly" gives us the "Substitution" in general scenarios.

We can check that the 2 equations will not let $x=0$ ( it will give $0=18$ ) & $y=0$ ( it will give $0=26$ )
Hence $\tan(\theta)=y/x$ is neither indeterminate nor $0$ nor $\infty$ here.
More-over , it is a real number in general , though it is a rational number in the given context.

Hence we take the Single Solution to the Cubic Equation which is the real number $1/3$ which is rational number too , according to the given context.
It means $z=3y+yi$ where $y$ is Non-Zero Integer.
We have to use this to get the overall Solution here.

When we use $z=3y+yi$ with $z^3=18+26i$ , we will get 3 Solutions , including the wanted Solution $z=3+1i$ , where the 2 Solutions will involve unwanted complicated values.