Question

Which linear equation has the same solution set as 9d/4+3−2d=−2(3+d) ?(1 point)
A. 3x/5 - 10 = - 7
B. 35 - 15b = 95
C. 14 = 7/4x - 19/4 +2x
D. 4w - 2 (7 - 3w) = w/3 - 6

Answer 1

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

Let's start by simplifying \( \frac{9d}{4} + 3 - 2d = -2(3 + d) \).

1. Distributing the right side:
\[
-2(3 + d) = -6 - 2d
\]
Thus, we rewrite the equation:
\[
\frac{9d}{4} + 3 - 2d = -6 - 2d
\]

2. Next, we can cancel the \( -2d \) from both sides (since it's equal on both sides, it won't change the solution):
\[
\frac{9d}{4} + 3 = -6
\]

3. Subtracting 3 from both sides gives:
\[
\frac{9d}{4} = -6 - 3
\]
\[
\frac{9d}{4} = -9
\]

4. Now, we can multiply both sides by 4 to eliminate the fraction:
\[
9d = -36
\]

5. Dividing both sides by 9 gives:
\[
d = -4
\]

Now, let's analyze the provided options to find out which one has the same solution of \( d = -4 \).

**Option A: \( \frac{3x}{5} - 10 = -7 \)**
- Solving this:
\[
\frac{3x}{5} = -7 + 10
\]
\[
\frac{3x}{5} = 3
\]
\[
3x = 15
\]
\[
x = 5
\]

**Option B: \( 35 - 15b = 95 \)**
- Solving this:
\[
-15b = 95 - 35
\]
\[
-15b = 60
\]
\[
b = -4
\]

**Option C: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)**
- To solve it, first combine like terms:
\[
14 = \frac{7}{4}x + \frac{8}{4}x - \frac{19}{4}
\]
\[
14 = \frac{15}{4}x - \frac{19}{4}
\]
Now add \( \frac{19}{4} \) to both sides:
\[
14 + \frac{19}{4} = \frac{15}{4}x
\]
Convert 14 to a fraction:
\[
\frac{56}{4} + \frac{19}{4} = \frac{15}{4}x
\]
\[
\frac{75}{4} = \frac{15}{4}x
\]
\[
75 = 15x
\]
\[
x = 5
\]

**Option D: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)**
- Expanding:
\[
4w - 14 + 6w = \frac{w}{3} - 6
\]
\[
10w - 14 = \frac{w}{3} - 6
\]
Multiply everything by 3 to eliminate the fraction:
\[
30w - 42 = w - 18
\]
\[
30w - w = 42 - 18
\]
\[
29w = 24
\]
\[
w = \frac{24}{29}
\]

The only equation that has the solution \( -4 \) is **Option B: \( 35 - 15b = 95 \)**. Thus, the correct answer is:

**B. 35 - 15b = 95**