To solve the problem, let's define the situation with equations based on the information given:
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The first machine has already filled 180 cans and continues to fill cans at a rate of 15 cans per minute. Therefore, after \( x \) minutes, the total number of cans filled by this machine will be: \[ 180 + 15x \]
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The second machine has already filled 210 cans and continues to fill cans at a rate of 12 cans per minute. After \( x \) minutes, the total number of cans filled by this machine will be: \[ 210 + 12x \]
Now, we want to find out when both machines have filled the same number of cans, which leads us to set the equations equal to each other: \[ 180 + 15x = 210 + 12x \]
Moving the terms around doesn't exactly match any of the responses directly, but we can manipulate the equation to fit one of the proposed equations.
Subtracting \( 12x \) from both sides: \[ 180 + 3x = 210 \]
Then, subtracting 180 from both sides results in: \[ 3x = 30 \]
Now, the equation can be transformed to fit the given response options. Rearranging this might return us to an appropriate option:
Let’s check the provided options against our derived equation:
- The first option \( 210 - 12x = 180 - 15x \) can be rearranged: To standard form:
- Start with: \( 210 - 12x = 180 - 15x \)
- Rearranging gives: \( 15x - 12x = 210 - 180 \)
- We get \( 3x = 30 \)
- Which confirms that our original calculation aligns with this rearranged form.
Thus, the correct choice is: 210 - 12x = 180 - 15x