To translate the given information into a system of equations, let's define the two numbers as \( x \) and \( y \).
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The difference of the two numbers is -6, which can be represented as: \[ x - y = -6 \]
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The sum of twice one of the numbers and the other number is 78 can be represented as: \[ 2x + y = 78 \]
Putting this together, the system of equations is: \[ x - y = -6 \quad \text{and} \quad 2x + y = 78 \]
The correct response is: x - y = -6 and 2x + y = 78.