To find out when the two machines will have filled the same number of cans of soup, we can set up the following equations based on the information provided:
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The first machine has filled 180 cans and fills cans at a rate of 15 cans per minute. Therefore, after \( x \) minutes, the total number of cans filled by the first machine will be: \[ 180 + 15x \]
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The second machine has filled 210 cans and fills cans at a rate of 12 cans per minute. Therefore, after \( x \) minutes, the total number of cans filled by the second machine will be: \[ 210 + 12x \]
To find out when the two machines have filled the same number of cans, we can set these two expressions equal to each other: \[ 180 + 15x = 210 + 12x \]
Thus, the equation that represents the situation is: \[ 180 + 15x = 210 + 12x \]
Now, let’s analyze the provided options with respect to this equation:
- 12x + 210 = 15x + 180: This is not the correct equation derived from the conditions.
- 12x + 180 = 15x + 210: This is not the correct equation as well.
- 210 − 12x = 180 − 15x: This is just rearranging the terms, but it leads to the same relationship. However, it’s not in the standard form we derived.
- 12x − 210 = 15x − 180: Also doesn't represent the situation correctly.
The correct format of the equation is as follows: \[ 180 + 15x = 210 + 12x \]
However, if we simply wanted to rearrange it, the closest version in the responses that resembles the correct form is: \[ 12x + 210 = 15x + 180 \] Therefore, the correct initial equation to state that the two machines will have filled the same number of cans of soup is: \[ \text{12x + 210 = 15x + 180} \]