The first vat is filling at a rate of 6 gallons per minute, so after x minutes, it will have x * 6 gallons of liquid.
The second vat is already filled with 10 gallons and filling at a rate of 4 gallons per minute, so after x minutes, it will have 10 + x * 4 gallons of liquid.
To find the time when the two vats will have an equal amount of liquid, we can set these two expressions equal to each other:
x * 6 = 10 + x * 4
Simplifying the equation:
6x = 10 + 4x
Subtracting 4x from both sides:
6x - 4x = 10
2x = 10
Dividing both sides by 2:
x = 5
Therefore, the two vats will have an equal amount of liquid after 5 minutes.