To determine which linear equation has the same solution set as the equation \( \frac{9d}{4} + 3 - 2d = -2(3 + d) \), we first need to simplify the given equation.
Starting from: \[ \frac{9d}{4} + 3 - 2d = -2(3 + d) \]
We can distribute on the right-hand side: \[ \frac{9d}{4} + 3 - 2d = -6 - 2d \]
Now, notice that on both sides we have \(-2d\), so we can eliminate \(-2d\) from both sides: \[ \frac{9d}{4} + 3 = -6 \]
Next, we subtract 3 from both sides: \[ \frac{9d}{4} = -6 - 3 \] \[ \frac{9d}{4} = -9 \]
Now, we will multiply both sides by 4 to eliminate the fraction: \[ 9d = -36 \]
Finally, we divide by 9 to solve for \(d\): \[ d = -4 \]
Now that we have determined that the original equation has a solution of \(d = -4\), we need to check which of the choices also equals \(d = -4\).
Checking Each Option:
A) \( 35 - 15b = 95 \) \[ -15b = 95 - 35 \] \[ -15b = 60 \] \[ b = -4 \]
This implies that \(b = -4\).
B) \( 14 = \frac{7}{4} x - \frac{19}{4} + 2x \) \[ 14 = \frac{7}{4} x - \frac{19}{4} + \frac{8}{4} x \] \[ 14 = \left( \frac{7}{4} + \frac{8}{4} \right)x - \frac{19}{4} \] \[ 14 = \frac{15}{4} x - \frac{19}{4} \] Multiply everything by 4: \[ 56 = 15x - 19 \] \[ 15x = 56 + 19 \] \[ 15x = 75 \implies x = 5 \]
This means \(x \neq -4\).
C) \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \) \[ 4w - 14 + 6w = \frac{w}{3} - 6 \] \[ 10w - 14 = \frac{w}{3} - 6 \] Multiply everything by 3 to eliminate fractions: \[ 30w - 42 = w - 18 \] \[ 29w = 24 \implies w = \frac{24}{29} \]
This means \(w \neq -4\).
D) \( \frac{3x}{5} - 10 = -7 \) \[ \frac{3x}{5} = 3 \] \[ 3x = 15 \implies x = 5 \] This means \(x \neq -4\).
Conclusion:
The only equation from the options that has the same solution of \(-4\) corresponds to \(b = -4\) in option A.
Thus, the correct answer is A) \(35 - 15b = 95\).
2 answers