Let's break down the word problem and translate it into equations step by step.
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"Six times a larger number is equal to the sum of a smaller number and 18."
- Let \( y \) be the larger number and \( x \) be the smaller number.
- This translates to the equation: \[ 6y = x + 18 \]
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"The difference of twice the larger number and the smaller number is 6."
- This translates to the equation: \[ 2y - x = 6 \]
Now, we will rewrite each equation in a more standard form:
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From \( 6y = x + 18 \):
- You could rearrange it as: \[ y = \frac{x + 18}{6} \]
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From \( 2y - x = 6 \):
- Rearranging this gives: \[ 2y = x + 6 \quad \Rightarrow \quad y = \frac{x + 6}{2} \]
Now, let's evaluate the answer choices based on our equations.
None of the options presented match the derived equations exactly, but we can find the correct relationships as:
- The set of equations is: \[ 6y = x + 18 \quad \text{and} \quad 2y - x = 6 \]
When evaluating the choices given:
- Choice 1: \(y = 6 x + 18\). This is incorrect for the first equation (should be \(x + 18 = 6y\)).
- Choice 2: \(y = 2 x - 6\). This incorrectly states the second equation as \(y\).
- Choice 3: \(y = 6 (x + 18)\). This does not represent our equations.
- Choice 4: \(y = \frac{1}{6}x + 3\). This is not correct.
- Choice 5: \(y = \frac{1}{2}x + 6\). This is not correct.
After a careful review, it seems there is no correct answer among the provided options based on the equations we've derived.
You may need to check for any transcription error in the original provided options or refer back to adjusting your values based on the correct equations derived above.
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