Question

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

35−15b=95
35 minus 15 b equals 95

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

Answer 1

To solve this problem, we first need to simplify the given equation \( 9d^4 + 3 - 2d = -2(3 + d) \).

### Step 1: Simplify the Given Equation

Start by distributing on the right side:
\[
9d^4 + 3 - 2d = -6 - 2d
\]

Now, move \(-2d\) over to the left side (note they cancel out):
\[
9d^4 + 3 = -6
\]

Next, isolate the variable by adding 6 to both sides:
\[
9d^4 + 3 + 6 = 0
\]
\[
9d^4 + 9 = 0
\]

Now factor out the common term:
\[
9(d^4 + 1) = 0
\]

Since \(d^4 + 1\) does not equal zero for real numbers, we can say that this equation does not have real solutions.

### Step 2: Check the Given Options

Now we will evaluate if any of the suggested equations have a similar structure:

1. \(35 - 15b = 95\)
- Rearranging gives \( b = -4 \) which has real solutions.

2. \( \frac{3x}{5} - 10 = -7 \)
- Rearranging gives \( x = 15 \) which has real solutions.

3. \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)
- Rearranging can yield real solutions depending on \(w\).

4. \( 14 = 74x - 194 + 2x \)
- Rearranging gives \( x = \frac{208}{76} = \frac{52}{19} \), which has real solutions as well.

### Conclusion

The original equation \(9d^4 + 3 - 2d = -2(3+d)\) effectively has no real solution since the simplified equation \(9(d^4 + 1) = 0\) implies \(d^4 = -1\), which has no real solutions. Therefore, the linear equation that has the same solution set as this original equation would also have no real solutions, which none of the options have.

Hence, there is no equation in the choices that mirrors the structure of the original. If they all provide a solution set, then none of them would be the correct answer given the original's nature.