Although I am currently studying fractional linear transformations in complex analysis, I suddenly had the need to find the "solution" of a system of linear equations but I could not. Unless I have made some mistake in the algebra on the way, the system is as follows:
$ac' = a'c$
$ad \ ' +bc' = a'd + b'c$
$bd \ ' = b'd$
Observe from the first and last equation that we have equal ratios. So if I can show that $ab' = a'b$ then this should imply that the variables are multiples of each other by some fixed constant. I am suspecting that I am to use the second equation in some manner (hopefully not brute force using any of the other two) to obtain the desired equality.
Any help will be greatly appreciated!