Suppose a man's work rate in jobs per day is 1 job per M days, or 1/M jobs per day.

Suppose a woman's work rate in jobs per day is 1 jobs per W days, or 1/W jobs per day.

In 6 days 1 man can do (1/M)*6 = 6/M (fraction of a job)
In 6 days 1 woman can do (1/W)*6 = 6/W (fraction of a job)

Therefore:

In 6 days 6 man can do 6*6/M jobs or 36/M (fraction of a job).
In 6 days 5 women can do 5*6/W jobs or 30/W (fraction of a job).

So in those 6 days they did 36/M+30/W which equals 1 job:

So one equation is 36/M + 30/W = 1

Therefore

In 10 days 3 man can do 3*10/M jobs or 30/M (fraction of a job).
In 10 days 4 women can do 4*10/W jobs or 40/W (fraction of a job).

So in those 10 days they did 30/M+40/W which equals 1 job:

So another equation is 30/M + 40/W = 1

The system is



It makes a mess to clear of fractions, so don't!

Use elimination.  Multiply the first equation by 4 and the second
by -3 to cancel the W terms:







So 1 man can do the job in 54 days.

Substitute in 
              
              

Now we can clear of fractions:

              
              

So 1 woman can do the job in 90 days. 

In 1 day 9 men can do 9/M=9/54=1/6 (fraction of a job).
In 1 day 15 women can do 15/W=15/90=1/6 (fraction of a job).

So together in one day they can do 1/6+1/6 = 2/6 = 1/3 of a job.

So it will take them 3 days

Edwin