To find which linear equation has the same solution set as \( \frac{9d}{4} + 3 - 2d = -2(3 + d) \), we can start by simplifying the original equation.
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Distributing on the right side: \[ \frac{9d}{4} + 3 - 2d = -6 - 2d \]
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Adding \(2d\) to both sides to eliminate \(d\) on the right: \[ \frac{9d}{4} + 3 = -6 \]
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Subtracting \(3\) from both sides: \[ \frac{9d}{4} = -6 - 3 \] \[ \frac{9d}{4} = -9 \]
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To remove the fraction, multiply both sides by \(4\): \[ 9d = -36 \]
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Finally, divide by \(9\): \[ d = -4 \]
Now let's check the proposed equations:
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Equation 1: \( \frac{3x}{5} - 10 = -7 \) \[ \frac{3x}{5} = 3 \implies 3x = 15 \implies x = 5 \] (Not equivalent)
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Equation 2: \( 14 = \frac{7}{4} x - \frac{19}{4} + 2x \)
- Combine terms: \[ 14 = \frac{7}{4} x + \frac{8}{4} x - \frac{19}{4} \] \[ 14 + \frac{19}{4} = \frac{15}{4} x \] Convert \(14\) to fraction: \[ \frac{56}{4} + \frac{19}{4} = \frac{15}{4} x \implies \frac{75}{4} = \frac{15}{4} x \implies x = 5 \] (Not equivalent)
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Equation 3: \( 35 - 15b = 95 \) \[ -15b = 95 - 35 \implies -15b = 60 \implies b = -4 \] (Equivalent)
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Equation 4: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \) \[ 4w - 14 + 6w = \frac{w}{3} - 6 \] \[ 10w - 8 = \frac{w}{3} - 6 \] Multiply the whole equation by 3: \[ 30w - 24 = w - 18 \] \[ 29w = 6 \implies w = \frac{6}{29} \] (Not equivalent)
The equation that has the same solution set as \(d = -4\) is \(35 - 15b = 95\).