To find a linear equation that has the same solution set as the given equation \(\frac {9d}{4}+3-2d=-2(3+d)\), we first need to simplify it.
Starting with the left side, we can combine terms:
-
Rewrite \(-2(3+d)\) on the right side: \[ -2(3+d) = -6 - 2d \]
This gives us: \[ \frac{9d}{4} + 3 - 2d = -6 - 2d \]
-
Add \(2d\) to both sides: \[ \frac{9d}{4} + 3 = -6 \]
-
Now, subtract 3 from both sides: \[ \frac{9d}{4} = -6 - 3 \] Which simplifies to: \[ \frac{9d}{4} = -9 \]
-
Multiply both sides by \(4\) to eliminate the fraction: \[ 9d = -36 \]
-
Finally, divide by \(9\) to solve for \(d\): \[ d = -4 \]
Now we need to check each of the following available equations to determine which has the same solution \(d = -4\):
-
Equating the first option \(4w - 2(7 - 3w) = \frac{w}{3} - 6\): \[ 4w - 14 + 6w = \frac{w}{3} - 6 \] Combine and simplify: \[ 10w - 14 = \frac{w}{3} - 6 \] Multiply everything by \(3\): \[ 30w - 42 = w - 18 \] Rearranging gives: \[ 29w = 24 \Rightarrow w = \frac{24}{29} \] Not the same solution.
-
For the second option \(35 - 15b = 95\): \[ -15b = 95 - 35 \] Simplifying: \[ -15b = 60 \Rightarrow b = -4 \] This is the same solution.
-
Next, check \(\frac {3x}{5} - 10 = -7\): \[ \frac{3x}{5} = 3 \Rightarrow 3x = 15 \Rightarrow x = 5 \] Not the same solution.
-
Finally, check \(14 = \frac{7}{4}x - \frac{19}{4} + 2x\): \[ \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x = 14 \] Combine terms: \[ \frac{15}{4}x - \frac{11}{4} = 14 \] Rearranging gives: \[ \frac{15}{4}x = 14 + \frac{11}{4} \] Calculate the right side: \[ 14 = \frac{56}{4} \Rightarrow \frac{15}{4}x = \frac{56 + 11}{4} = \frac{67}{4} \] This simplifies to: \[ 15x = 67 \Rightarrow x = \frac{67}{15} \] Not the same solution.
After checking all options, the only equation that has the same solution set as the original equation is:
\[ \boxed{35 - 15b = 95} \]
1 answer