The overall equation is y = x. This means the output is the same as the input.
The number machine first multiplies x by 6, giving 6x.
To get from 6x back to x, we must perform the inverse operation of multiplying by 6.
The inverse of multiplying by 6 is dividing by 6.
So, the second box must be "÷ 6".
Check: (x × 6) ÷ 6 = 6x ÷ 6 = x. This is correct.
Full Answer
The second box should be "÷ 6".
The number machine must result in the output `y` being equal to the input `x`.
Let's trace the operations:
1. The input `x` is first multiplied by 6. The result is `6x`.
2. This result `6x` then goes through a second operation in the empty box to produce the final output `y`.
We know that the final output must be `x`. So, we have the equation:
(6x) [box] = x
We need to find an operation that turns `6x` into `x`. To do this, we must divide by 6.
So, the operation in the box is **÷ 6**.
This is because multiplication and division are inverse operations. If you multiply by a number and then divide by the same number, you get back to where you started.
Common mistakes
✗ Putting "- 6" in the box. This would give 6x - 6, which is not equal to x.\n✗ Putting "× (1/6)" is correct mathematically but "÷ 6" is the more standard answer.
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