Question
which linear equation has the same solution sets as 9d/4 + 3 = -2 (3+d)
a. 3x/5 -1 10 = -7
b. 35 - 15bb = 95
c. 14 = 7/4x - 19/4 + 2x
d. 4w - 2 (7 - 3w) = w/3 - 6
a. 3x/5 -1 10 = -7
b. 35 - 15bb = 95
c. 14 = 7/4x - 19/4 + 2x
d. 4w - 2 (7 - 3w) = w/3 - 6
Answers
GPT-4o mini
To determine which linear equation has the same solution set as the given equation \( \frac{9d}{4} + 3 = -2(3 + d) \), we will solve the original equation for \( d \) and then check each option.
Starting with the equation:
\[
\frac{9d}{4} + 3 = -2(3 + d)
\]
First, distribute \(-2\) on the right side:
\[
\frac{9d}{4} + 3 = -6 - 2d
\]
Next, simplify this by bringing \(2d\) to the left side and \(3\) to the right side:
\[
\frac{9d}{4} + 2d = -6 - 3
\]
\[
\frac{9d}{4} + \frac{8d}{4} = -9
\]
Combine the terms:
\[
\frac{17d}{4} = -9
\]
Now, multiply both sides by \(4\) to eliminate the fraction:
\[
17d = -36
\]
Now, isolate \(d\):
\[
d = -\frac{36}{17}
\]
Now, we will analyze each option to see if it has the same solution, \(d = -\frac{36}{17}\).
**Option A: \( \frac{3x}{5} - 10 = -7 \)**
Rearranging:
\[
\frac{3x}{5} = 3 \implies 3x = 15 \implies x = 5
\]
**Not equal to \( -\frac{36}{17}\)**.
**Option B: \( 35 - 15bb = 95 \)**
Rearranging this gives:
\[
-15bb = 95 - 35 \implies -15bb = 60 \implies bb = -4
\]
**Not equal to \( -\frac{36}{17}\)**. \(b\) would be imaginary.
**Option C: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)**
Rearranging gives:
\[
14 = \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x
\]
Combining like terms:
\[
14 = \frac{15}{4}x - \frac{11}{4}
\]
Rearranging leads to:
\[
\frac{15}{4}x = 14 + \frac{11}{4}
\]
Converting \(14\) into fourths:
\[
14 = \frac{56}{4} \Rightarrow \frac{15}{4}x = \frac{56 + 11}{4} = \frac{67}{4}
\]
Multiply both sides by \( \frac{4}{15} \):
\[
x = \frac{67}{15} \implies x = 4.47 \text{ but not equal to } -\frac{36}{17}
\]
**Option D: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)**
Expanding:
\[
4w - 14 + 6w = \frac{w}{3} - 6
\]
Combine:
\[
10w - 14 = \frac{w}{3} - 6
\]
Multiply through by \(3\) to eliminate fractions:
\[
30w - 42 = w - 18
\]
Simplifying gives:
\[
29w = 24 \implies w = \frac{24}{29}
\]
**Result**:
None of the provided equations has the same solution set as the original equation \(d = -\frac{36}{17}\).
Starting with the equation:
\[
\frac{9d}{4} + 3 = -2(3 + d)
\]
First, distribute \(-2\) on the right side:
\[
\frac{9d}{4} + 3 = -6 - 2d
\]
Next, simplify this by bringing \(2d\) to the left side and \(3\) to the right side:
\[
\frac{9d}{4} + 2d = -6 - 3
\]
\[
\frac{9d}{4} + \frac{8d}{4} = -9
\]
Combine the terms:
\[
\frac{17d}{4} = -9
\]
Now, multiply both sides by \(4\) to eliminate the fraction:
\[
17d = -36
\]
Now, isolate \(d\):
\[
d = -\frac{36}{17}
\]
Now, we will analyze each option to see if it has the same solution, \(d = -\frac{36}{17}\).
**Option A: \( \frac{3x}{5} - 10 = -7 \)**
Rearranging:
\[
\frac{3x}{5} = 3 \implies 3x = 15 \implies x = 5
\]
**Not equal to \( -\frac{36}{17}\)**.
**Option B: \( 35 - 15bb = 95 \)**
Rearranging this gives:
\[
-15bb = 95 - 35 \implies -15bb = 60 \implies bb = -4
\]
**Not equal to \( -\frac{36}{17}\)**. \(b\) would be imaginary.
**Option C: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)**
Rearranging gives:
\[
14 = \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x
\]
Combining like terms:
\[
14 = \frac{15}{4}x - \frac{11}{4}
\]
Rearranging leads to:
\[
\frac{15}{4}x = 14 + \frac{11}{4}
\]
Converting \(14\) into fourths:
\[
14 = \frac{56}{4} \Rightarrow \frac{15}{4}x = \frac{56 + 11}{4} = \frac{67}{4}
\]
Multiply both sides by \( \frac{4}{15} \):
\[
x = \frac{67}{15} \implies x = 4.47 \text{ but not equal to } -\frac{36}{17}
\]
**Option D: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)**
Expanding:
\[
4w - 14 + 6w = \frac{w}{3} - 6
\]
Combine:
\[
10w - 14 = \frac{w}{3} - 6
\]
Multiply through by \(3\) to eliminate fractions:
\[
30w - 42 = w - 18
\]
Simplifying gives:
\[
29w = 24 \implies w = \frac{24}{29}
\]
**Result**:
None of the provided equations has the same solution set as the original equation \(d = -\frac{36}{17}\).