Question

Which linear equation has the same solution set as $\frac {9d}{4}+3-2d=-2(3+d)$ ? (1 point)
$4w-2(7-3w)=\frac {w}{3}-6$
$35-15b=95$
$\frac {3x}{5}-10=-7$
$14=\frac {7}{4}x-\frac {19}{4}+2x$

Answer 1

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:

1. Rewrite \(-2(3+d)\) on the right side:
\[
-2(3+d) = -6 - 2d
\]

This gives us:
\[
\frac{9d}{4} + 3 - 2d = -6 - 2d
\]

2. Add \(2d\) to both sides:
\[
\frac{9d}{4} + 3 = -6
\]

3. Now, subtract 3 from both sides:
\[
\frac{9d}{4} = -6 - 3
\]
Which simplifies to:
\[
\frac{9d}{4} = -9
\]

4. Multiply both sides by \(4\) to eliminate the fraction:
\[
9d = -36
\]

5. 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 \(d = -4\):

1. **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.

2. **For the second option \(35 - 15b = 95\)**:
\[
-15b = 95 - 35
\]
Simplifying:
\[
-15b = 60 \Rightarrow b = -4
\]
This is the same solution.

3. **Next, check \(\frac {3x}{5} - 10 = -7\)**:
\[
\frac{3x}{5} = 3 \Rightarrow 3x = 15 \Rightarrow x = 5
\]
Not the same solution.

4. **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:

\[
\boxed{35 - 15b = 95}
\]