Rates of more than one eater at the same time are additive.

j, time for Jenny
p, time for Penny
y, time for Lenny
To eat the one gallon on each's own



Notice that the numerators in all terms are 1, and only a variable is used in every denominator on the left hand members.
An alternative is to not to focus on the times as variables, but to focus on the RATES directly as variables, 
although each rate will still be a fraction.

You might be more comfortable reassigning new variables J = , P =  and Y =  and forming

J + P     = ,     (1)    
P +     Y = ,     (2)          ( <----- 3/5 =  )
J + P + Y = .     (3)

Now you have a system of LINEAR equations instead of RATIONAL equations. 

Add equations (1) and (2). You will get

J + 2P + Y =  =  = .   (4)

Now distract equation (3) from (4). You will get

P =  = .   (You just have the rate for Lenny!)

Next, from (1) you get J =  - P =  =  = .

This is Jenny's rate.

Hence, it will take  =  =  =  hours for Jenny to eat a whole gallon of ice cream by herself.

The problem is solved.

Here's the easiest way to solve the problem.  No algebra needed.
Just basic math.  Not only that but also too much information is 
given.  We don't even need the first sentence at all.

The times given in the other sentences are 1 2/3 hours, and 
1 hour.

1 2/3 hours is 1 hour 40 minutes or 100 minutes.

So the times given are 100 minutes and 60 minutes.

The easiest common multiple of those is 600 minutes or 10 hours.
So let's take everything up to 10 hours.

That's 100 minutes, so in 600 minutes (or 10 hours) Penney 
and Lenny can eat 6 gallons. 

So in 10 hours they can all three eat 10 gallons.

Since in 10 hours Penney and Lenny would eat 6 of those 10 
gallons, Jenny would eat the other 4.  Jenny can eat 4 
gallons in 10 hours, so Jenny could eat 1 gallon in 1/4th 
of 10 hours or 2 1/2 hours.  

See?  We didn't need the first sentence, or any algebra, 
at all. :)

Edwin