To solve the expression \( \frac{2(17 + 2x)}{y^2} - 11 \) when \( x = 4 \) and \( y = 6 \), we can follow these steps:
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Substitute the values of \( x \) and \( y \) into the expression.
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Calculate \( y^2 \): \[ y^2 = 6^2 = 36 \]
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Substitute \( x = 4 \) into \( 17 + 2x \): \[ 17 + 2(4) = 17 + 8 = 25 \]
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Now, substitute these results into the expression: \[ \frac{2(25)}{36} - 11 \]
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Calculate \( 2(25) \): \[ 2(25) = 50 \]
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Now, substitute back into the expression: \[ \frac{50}{36} - 11 \]
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Simplify \( \frac{50}{36} \): \[ \frac{50}{36} = \frac{25}{18} \]
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Therefore, the expression now is: \[ \frac{25}{18} - 11 \]
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Convert \( 11 \) to a fraction with a denominator of 18: \[ 11 = \frac{198}{18} \]
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Now perform the subtraction: \[ \frac{25}{18} - \frac{198}{18} = \frac{25 - 198}{18} = \frac{-173}{18} \]
Thus, the final result is: \[ \frac{-173}{18} \]