A 19 liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y liters of water are added to this mixture to make a 54 liter mixture consisting by volume of 1 part juice to 2 parts water, then what is the value of x?

Let's break down the problem step by step:

1. We have a 19-liter mixture consisting of 1 part juice to 18 parts water. This means that in the initial mixture, there is 1/19 of juice and 18/19 of water.
  - Juice in the initial mixture = (1/19) * 19 liters = 1 liter
  - Water in the initial mixture = (18/19) * 19 liters = 18 liters

2. We are adding x liters of juice and y liters of water to this mixture to make a 54-liter mixture consisting of 1 part juice to 2 parts water.
  - Juice in the final mixture = 1/3 of the total volume = (1/3) * 54 liters = 18 liters
  - Water in the final mixture = 2/3 of the total volume = (2/3) * 54 liters = 36 liters

3. Since we started with 1 liter of juice and added x liters of juice, the total juice in the final mixture is 1 + x liters.

4. We can set up the equation:
  Juice in the final mixture = Juice in the initial mixture + Juice added
  18 liters = 1 liter + x liters
  18 = 1 + x

Now we can solve the equation to find the value of x:

18 = 1 + x
x = 18 - 1
x = 17

Therefore, the value of x is 17 liters.