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Which linear equation has the same solution set as 9d4+3−2d=−2(3+d)?(1 point)

Responses

3x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7

4w−2(7−3w)=w3−6
4 w minus 2 left parenthesis 7 minus 3 w right parenthesis equals Start Fraction w over 3 End Fraction minus 6

14=74x−194+2x
14 equals Start Fraction 7 over 4 End Fraction x minus Start Fraction 19 over 4 End Fraction plus 2 x

35−15b=95

3 answers

To determine which linear equation has the same solution set as \( 9d^4 + 3 - 2d = -2(3 + d) \), we first need to simplify the given equation.

  1. Distribute on the right side: \[ 9d^4 + 3 - 2d = -2(3 + d) = -6 - 2d \]

    So our equation is: \[ 9d^4 + 3 - 2d = -6 - 2d \]

  2. Notice that \(-2d\) cancels from both sides: \[ 9d^4 + 3 = -6 \]

  3. Now, move \(3\) to the right: \[ 9d^4 = -6 - 3 \] \[ 9d^4 = -9 \]

  4. Divide by \(9\): \[ d^4 = -1 \]

The equation \(d^4 = -1\) has no real solutions because no real number raised to the fourth power can yield a negative result.

Now, let’s analyze each response to see which has no real solutions:

  1. \(3x/5 - 10 = -7\): Solving this gives: \[ 3x/5 = 3 \implies x = 5 \] (Has real solution)

  2. \(4w - 2(7 - 3w) = w/3 - 6\): Simplifying gives: \[ 4w - 14 + 6w = w/3 - 6 \] Combine like terms, and you’ll likely find a real solution.

  3. \(14 = 7/4 x - 19/4 + 2x\): Rearranging this will also yield a real solution.

  4. \(35 - 15b = 95\): Solving gives: \[ -15b = 60 \implies b = -4 \] (Has real solution)

None of the provided equations appear to fit the conclusion that they would have no real solutions like \(d^4 = -1\). Therefore, the initial equation does not seem to have a directly corresponding option with the same characteristics.

Thus, none of the response options given possess the same characteristics as the original equation \(9d^4 + 3 - 2d = -2(3 + d)\). However, you might want to critically evaluate the other signs of "equivalent" if there are special conditions or if the question is asking for context outside of real solutions.

Pick the answer that is most right

Let's evaluate each equation again to determine if any might be considered "most right" in terms of having the same solution characteristics (specifically, no real solutions):

  1. \( \frac{3x}{5} - 10 = -7 \): \[ \frac{3x}{5} = 3 \implies 3x = 15 \implies x = 5 \] (Has a real solution.)

  2. \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \): This simplifies to: \[ 4w - 14 + 6w = \frac{w}{3} - 6 \implies 10w - 8 = \frac{w}{3} - 6 \] Solving this will yield a real solution.

  3. \( 14 = \frac{7}{4} x - \frac{19}{4} + 2x \): Solving this: \[ 14 + \frac{19}{4} = \frac{7}{4}x + 2x \implies \frac{14}{1} + \frac{19}{4} \text{ will yield a real x.} \]

  4. \( 35 - 15b = 95 \): \[ -15b = 60 \implies b = -4 \] (Has a real solution.)

Upon examining all options, none yield no real solutions directly. However, the equation which may diverge from typical real solutions might be:

Option 2 interpretable for complex solutions:

\( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \), could lead to a complex solution but on surface calculation may not seem like getting a negative outcome to fit basis as well.

Given the nature of the requirement that closely aligns itself to a fourth-degree polynomial regarding real solutions:

Final Pick:

In light of seeking something that might yield complications and negative square roots leading to non-real results, it could have indicate:

The best answer is (based purely on examination above): \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \).

It hints potentially towards complex outcomes though further adjustments required for satisfactory relation to primary conditions. However, context indicates none meet the solvability without real solutions distinctly as noted for the original equation.