Question
Which linear equation has the same solution set as 9d/4 +3−2d=−2(3+d)?(1 point)
Responses
3x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
14=74x−194+2x
14 equals Start Fraction 7 over 4 End Fraction x minus Start Fraction 19 over 4 End Fraction plus 2 x
4w−2(7−3w)=w3−6
4 w minus 2 left parenthesis 7 minus 3 w right parenthesis equals Start Fraction w over 3 End Fraction minus 6
35−15b=95
35 minus 15 b equals 95
Responses
3x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
14=74x−194+2x
14 equals Start Fraction 7 over 4 End Fraction x minus Start Fraction 19 over 4 End Fraction plus 2 x
4w−2(7−3w)=w3−6
4 w minus 2 left parenthesis 7 minus 3 w right parenthesis equals Start Fraction w over 3 End Fraction minus 6
35−15b=95
35 minus 15 b equals 95
Answers
GPT-4o mini
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 left-hand side and the right-hand side.
1. Start with the equation:
\[
\frac{9d}{4} + 3 - 2d = -2(3 + d)
\]
2. Simplify the right-hand side:
\[
-2(3 + d) = -6 - 2d
\]
3. Rewrite the equation:
\[
\frac{9d}{4} + 3 - 2d = -6 - 2d
\]
4. Notice that \( -2d \) can be subtracted from both sides:
\[
\frac{9d}{4} + 3 = -6
\]
5. Now, isolate \( \frac{9d}{4} \):
\[
\frac{9d}{4} = -6 - 3
\]
\[
\frac{9d}{4} = -9
\]
6. Multiply both sides by 4 to eliminate the fraction:
\[
9d = -36
\]
7. Divide by 9:
\[
d = -4
\]
Now we have \( d = -4 \). We want to find which of the equations given has the same solution set.
Next, we will evaluate each option to check if \( d = -4 \) is a solution:
1. **Option 1:** \( \frac{3x}{5} - 10 = -7 \)
\( \frac{3x}{5} = 3 \)
\( 3x = 15 \)
\( x = 5 \)
2. **Option 2:** \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)
Multiply through by 4:
\( 56 = 7x - 19 + 8x \)
\( 56 = 15x - 19 \)
\( 75 = 15x \)
\( x = 5 \)
3. **Option 3:** \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)
\( 4w - 14 + 6w = \frac{w}{3} - 6 \)
\( 10w - 14 = \frac{w}{3} - 6 \)
Multiply through by 3:
\( 30w - 42 = w - 18 \)
\( 29w = 24 \)
\( w = \frac{24}{29} \)
4. **Option 4:** \( 35 - 15b = 95 \)
\( -15b = 60 \)
\( b = -4 \)
The original equation has a solution \( d = -4 \), and the only option that yields the same solution is:
**35 - 15b = 95.**
1. Start with the equation:
\[
\frac{9d}{4} + 3 - 2d = -2(3 + d)
\]
2. Simplify the right-hand side:
\[
-2(3 + d) = -6 - 2d
\]
3. Rewrite the equation:
\[
\frac{9d}{4} + 3 - 2d = -6 - 2d
\]
4. Notice that \( -2d \) can be subtracted from both sides:
\[
\frac{9d}{4} + 3 = -6
\]
5. Now, isolate \( \frac{9d}{4} \):
\[
\frac{9d}{4} = -6 - 3
\]
\[
\frac{9d}{4} = -9
\]
6. Multiply both sides by 4 to eliminate the fraction:
\[
9d = -36
\]
7. Divide by 9:
\[
d = -4
\]
Now we have \( d = -4 \). We want to find which of the equations given has the same solution set.
Next, we will evaluate each option to check if \( d = -4 \) is a solution:
1. **Option 1:** \( \frac{3x}{5} - 10 = -7 \)
\( \frac{3x}{5} = 3 \)
\( 3x = 15 \)
\( x = 5 \)
2. **Option 2:** \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)
Multiply through by 4:
\( 56 = 7x - 19 + 8x \)
\( 56 = 15x - 19 \)
\( 75 = 15x \)
\( x = 5 \)
3. **Option 3:** \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)
\( 4w - 14 + 6w = \frac{w}{3} - 6 \)
\( 10w - 14 = \frac{w}{3} - 6 \)
Multiply through by 3:
\( 30w - 42 = w - 18 \)
\( 29w = 24 \)
\( w = \frac{24}{29} \)
4. **Option 4:** \( 35 - 15b = 95 \)
\( -15b = 60 \)
\( b = -4 \)
The original equation has a solution \( d = -4 \), and the only option that yields the same solution is:
**35 - 15b = 95.**