special relativity - Why do we write the lengths in the following way? Question about Lorentz transformation

Yesterday we have studied the Lorentz transformation in school. So we have two frames of reference, $S$ and $S'$ . $S$ is stationary and $S'$. $S'$ has a constant velocity $v$, relative to the $S$ frame. $v$ is directed along the Ox axis. Ox is parallel to Ox' and Oy is parallel to Oy'.

If we apply the Galilran Transformations we get:

$x = x' + ut'$ $y = y'$ $z = z'$ $t = t'$

$x' = x - ut$ $y'=y$ $z'=z$ $t' = t$

Now, our physics teacher, assumed that:

$x=k(x'+ut')$ $x'=k(x-ut)$ with k being a constant.

Why did he do that? I didn't understand. I undrstood that the length of an object depends o the frame of reference and that the speed of light is the same in the two frames.

Assuming the above facts, we can derive the $k$ constant:

$$\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$$

But why did we make that first assumptikn? I didn't get the logic. Could somebody explain, please?