I think that in order to have a good discussion it would help if you could explain why public transport is slower in your case.
Is the route longer because there isn't a line that goes straight from your home to your workplace?
Is it because there aren't a lot of trains, so they take too long to arrive?
Otherwise it's hard to tell what the problem with public transport in your city.
I live in London for instance, where cars are _almost always_ the slowest option for trips around zones 1-2 or even 3, because the center is always very busy. So when commuting here the fastest option is usually cycling, then 5-10 minutes slower is public transport and driving, which depending on the time of day will also be slower...
Public transit is only faster than cars when they have an "unfair" advantage. I'm guessing (I haven't been there) by London zone 1-2 you means trips where the subway (tube you call it?) gets transit out of the way. Other useful "unfair" advantage are trains - because they cannot stop - preempt traffic, and there are also transit only lanes where cars are not allowed. Anytime transit doesn't get some advantage over traffic it is slower because it has the same traffic plus the additional non-traffic stops it needs to make. For long distance trips transit is often faster.
The unfair advantage is not cost effective to grant to all cars, but still need to be acknowledged.
In a city where the availability of land is a major limiting factor, you could argue that cars are the ones getting an "unfair advantage" - 8 square meters of city surface permanently dedicated to moving oneself around is clearly an inequitable use of space.
Is the route longer because there isn't a line that goes straight from your home to your workplace?
Is it because there aren't a lot of trains, so they take too long to arrive?
Otherwise it's hard to tell what the problem with public transport in your city.
I live in London for instance, where cars are _almost always_ the slowest option for trips around zones 1-2 or even 3, because the center is always very busy. So when commuting here the fastest option is usually cycling, then 5-10 minutes slower is public transport and driving, which depending on the time of day will also be slower...