To find which linear equation has the same solution set as the given equation \( 9d^4 + 3 - 2d = -2(3 + d) \), we can first simplify the equation step-by-step.
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Distribute the \(-2\) on the right side: \[ 9d^4 + 3 - 2d = -6 - 2d \]
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Add \(2d\) to both sides: \[ 9d^4 + 3 = -6 \]
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Now, add \(6\) to both sides: \[ 9d^4 + 3 + 6 = 0 \] Which simplifies to: \[ 9d^4 + 9 = 0 \]
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Factor or rearrange the equation: \[ 9(d^4 + 1) = 0 \] This implies \(d^4 + 1 = 0\), which does not yield any real solutions.
The next step involves analyzing each provided linear equation to see if any can be equated or transformed into a similar solution framework.
Let's go through each option to identify its nature and whether any of them simplifies to a similar non-real condition:
- Option 1: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)
- Option 2: \( 35 - 15b = 95 \)
- Option 3: \( \frac{3x}{5} - 10 = -7 \)
- Option 4: \( 4w - 2(7 - 3w) = w^3 - 6 \)
To find the linear equations:
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Option 2 simplifies to: \[ 35 - 15b = 95 \implies -15b = 95 - 35 \implies -15b = 60 \implies b = -4 \] This gives a specific solution \(b = -4\).
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Option 3 simplifies to: \[ \frac{3x}{5} - 10 = -7 \implies \frac{3x}{5} = 3 \implies 3x = 15 \implies x = 5 \] This gives a specific solution \(x = 5\) as well.
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Option 4 simplifies to: \[ 4w - 14 + 6w = w^3 \implies 10w - 14 = w^3 \] This leads to a cubic equation which can yield more complicated solutions.
After analyzing the possibilities, the first option involves a linear combination of terms with fractions, potential to yield more extensive solutions or complexity.
Since you’re looking for an option that parallels the condition of no real solutions, none of these direct options appear to yield infinite or limitless solution sets as might be derived from \(d^4 + 1 = 0\). However, focusing on the complicated structure (possibly Option 4) provides a reminder of degrees beyond linearity rather than complete regression.
Therefore, since all considered options yield specific solutions or require further quartic review within the real field, none of the alternate equations match the characteristics of containing no real solutions like \(d^4 + 1 = 0\).
In conclusion, while you may certainly hold interest in complex solutions or looking for a potential filter to match non-linear outcomes, my alignment remains clear that it’s best to assert than any viable selection does not align similarly and requires clarity or dimensional modification.