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Two pumps are filling large vats of liquid. One vat is empty and the pump is filling it at a rate of 6 gallons per minute. The other is already filled with 10 gallons and it's continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have an equal amount of liquid. Use x as your variable

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Let \( x \) represent the number of minutes that have passed.

For the first vat that starts empty and fills at a rate of 6 gallons per minute, the amount of liquid in the first vat after \( x \) minutes can be expressed as:

\[ A_1 = 6x \]

For the second vat, which starts with 10 gallons and fills at a rate of 4 gallons per minute, the amount of liquid in the second vat after \( x \) minutes is:

\[ A_2 = 10 + 4x \]

To find out when the two vats will have an equal amount of liquid, we set \( A_1 \) equal to \( A_2 \):

\[ 6x = 10 + 4x \]

Now, we can solve for \( x \):

  1. Subtract \( 4x \) from both sides:

\[ 6x - 4x = 10 \]

  1. This simplifies to:

\[ 2x = 10 \]

  1. Now, divide both sides by 2:

\[ x = 5 \]

So, after 5 minutes, both vats will have an equal amount of liquid.