Are there tricks to solve complex understaffed linear systems by "guessing" a partial solution (first x=n*lambda with n!= 1)?

I solved a complex LGS problem from a workbook, but it took me ages and I had to experiment a lot and use a value from the given solution to even get there.
Now I'm wondering if I'm missing a trick or if there are no better solutions since I haven't found any better tips on the internet?

It was an understaffed LGS where all choices of x1,x2,x3,x4 = lambda only led to ugly fractions.
So you first had to find a solution for one of x1,x2,x3,x4 and in this task none of the choices were real but all were complex and far away from (1+i) or similar – rather numbers like (43+141i).

I have no idea how to come up with such a solution without at least looking at one solution and then experimenting with how to get this value through transformations and addition or subtraction from rows to others.

Unfortunately, I cannot enter the entire problem here, but my problem with complex LGS is to recognize which rows I have to calculate with which in order to get good values, because I regularly ended up with intermediate results after which the rows looked something like this:
34/135 + 173/135i until the final result yielded 4-6 digit numbers in the denominator. When I then tried to insert the solutions, this no longer produced the correct results, which I think must be due to rounding errors.

I have used all sorts of tricks, such as multiplying rows by reals, complex numbers and fractions, making the denominators real by multiplying by fraction*(conjugation/conjugation) and multiplying by i to change the order of the real and imaginary parts, but even when I thought I had done something clever, an impossible value would appear somewhere else in the row.
Maybe I just don't have an eye for complex numbers and fractions and how to recognize which factor to choose, but then there must be certain possibilities for that.

Can someone explain to me what options there are for checking LGS first in complex cases to determine which rows to choose and how to find the right factor for column multiplication to avoid ugly results?

I've created a sample exercise, but unfortunately, I'm not entirely sure if it has as good a solution as the exercises in my book… It's meant to serve more as an illustration:
(1) (1+i) (-4i) (5i) | (-5+29i)

(7) (3i) (5) (2-3i) | (15-13i)

(i) (0) (4) (i) | (-1+3i)
The solution would then look something like this:
L= {(2+i),(3i),(-i),(5)} + lambda*{(),(),(),()}
However, I'm not sure how to create the task so that everything is still correct behind the lambda, but in my task there were values ​​like 43+141i etc. behind the lambda.

Thank you!