Question
Which linear equation has the same solution set as 9d4+3−2d=−2(3+d)9d4+3−2d=−2(3+d)?(1 point)
Responses
35−15b=9535−15b=95
35 minus 15 b equals 95
4w−2(7−3w)=w3−64w−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
3x5−10=−73x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
14=74x−194+2x
Responses
35−15b=9535−15b=95
35 minus 15 b equals 95
4w−2(7−3w)=w3−64w−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
3x5−10=−73x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
14=74x−194+2x
Answers
GPT-4o mini
To determine which linear equation has the same solution set as the given equation \(9d + 3 - 2d = -2(3 + d)\), we will first simplify the given equation.
Starting with the equation:
\[
9d + 3 - 2d = -2(3 + d)
\]
First, simplify the left-hand side:
\[
(9d - 2d) + 3 = 7d + 3
\]
Now simplify the right-hand side:
\[
-2(3 + d) = -6 - 2d
\]
Now we rewrite the equation:
\[
7d + 3 = -6 - 2d
\]
Next, isolate \(d\) by moving all \(d\) terms to one side and constant terms to the other:
\[
7d + 2d = -6 - 3
\]
\[
9d = -9
\]
\[
d = -1
\]
So the solution to the original equation is \(d = -1\).
Now we need to check which of the provided equations also has \(d = -1\) as a solution:
1. **Response 1:** \(35 - 15b = 95\)
This can be simplified:
\[
-15b = 95 - 35 \Rightarrow -15b = 60 \Rightarrow b = -4
\]
This does not have \(d = -1\) as a solution.
2. **Response 2:** \(4w - 2(7 - 3w) = \frac{w}{3} - 6\)
Simplify the left-hand side:
\[
4w - 14 + 6w = 10w - 14
\]
The right-hand side becomes:
\[
\frac{w}{3} - 6
\]
This doesn't lead to an easy isolation of a variable; let's check it later if needed.
3. **Response 3:** \(\frac{3x}{5} - 10 = -7\)
This can be simplified:
\[
\frac{3x}{5} = -7 + 10 \Rightarrow \frac{3x}{5} = 3
\]
\[
3x = 15 \Rightarrow x = 5
\]
The solution here is \(x = 5\) which doesn't match.
4. **Response 4:** \(14 = 74x - 194 + 2x\)
Rearranging gives:
\[
14 = 76x - 194
\]
\[
76x = 14 + 194 \Rightarrow 76x = 208 \Rightarrow x = \frac{208}{76} = \frac{52}{19}
\]
This does not match.
After checking all equations, it seems we must analyze the second one carefully to ensure it provides a solution of \(d = -1\).
Revisiting:
**Response 2:** \(4w - 2(7 - 3w) = \frac{w}{3} - 6\)
Expanding the left-hand side gives:
\[
4w - 14 + 6w = 10w - 14
\]
Set it equal to \(\frac{w}{3} - 6\). Multiply every term by 3 to clear the fraction:
\[
30w - 42 = w - 18
\]
Now, bringing the \(w\) terms together yields:
\[
30w - w = 42 - 18
\]
\[
29w = 24 \Rightarrow w = \frac{24}{29}
\]
Not the correct solution.
**Iteratively correcting away** other calculations — None match the solution sets.
In conclusion, the direct verification upon numeric adjustments shows none of the obtained equations equate to \(d = -1\) originally except clearly observing \glossary into further contextual identities shall map into solving better without blind requests.
None of the equations in the provided responses have the same solution set.
Starting with the equation:
\[
9d + 3 - 2d = -2(3 + d)
\]
First, simplify the left-hand side:
\[
(9d - 2d) + 3 = 7d + 3
\]
Now simplify the right-hand side:
\[
-2(3 + d) = -6 - 2d
\]
Now we rewrite the equation:
\[
7d + 3 = -6 - 2d
\]
Next, isolate \(d\) by moving all \(d\) terms to one side and constant terms to the other:
\[
7d + 2d = -6 - 3
\]
\[
9d = -9
\]
\[
d = -1
\]
So the solution to the original equation is \(d = -1\).
Now we need to check which of the provided equations also has \(d = -1\) as a solution:
1. **Response 1:** \(35 - 15b = 95\)
This can be simplified:
\[
-15b = 95 - 35 \Rightarrow -15b = 60 \Rightarrow b = -4
\]
This does not have \(d = -1\) as a solution.
2. **Response 2:** \(4w - 2(7 - 3w) = \frac{w}{3} - 6\)
Simplify the left-hand side:
\[
4w - 14 + 6w = 10w - 14
\]
The right-hand side becomes:
\[
\frac{w}{3} - 6
\]
This doesn't lead to an easy isolation of a variable; let's check it later if needed.
3. **Response 3:** \(\frac{3x}{5} - 10 = -7\)
This can be simplified:
\[
\frac{3x}{5} = -7 + 10 \Rightarrow \frac{3x}{5} = 3
\]
\[
3x = 15 \Rightarrow x = 5
\]
The solution here is \(x = 5\) which doesn't match.
4. **Response 4:** \(14 = 74x - 194 + 2x\)
Rearranging gives:
\[
14 = 76x - 194
\]
\[
76x = 14 + 194 \Rightarrow 76x = 208 \Rightarrow x = \frac{208}{76} = \frac{52}{19}
\]
This does not match.
After checking all equations, it seems we must analyze the second one carefully to ensure it provides a solution of \(d = -1\).
Revisiting:
**Response 2:** \(4w - 2(7 - 3w) = \frac{w}{3} - 6\)
Expanding the left-hand side gives:
\[
4w - 14 + 6w = 10w - 14
\]
Set it equal to \(\frac{w}{3} - 6\). Multiply every term by 3 to clear the fraction:
\[
30w - 42 = w - 18
\]
Now, bringing the \(w\) terms together yields:
\[
30w - w = 42 - 18
\]
\[
29w = 24 \Rightarrow w = \frac{24}{29}
\]
Not the correct solution.
**Iteratively correcting away** other calculations — None match the solution sets.
In conclusion, the direct verification upon numeric adjustments shows none of the obtained equations equate to \(d = -1\) originally except clearly observing \glossary into further contextual identities shall map into solving better without blind requests.
None of the equations in the provided responses have the same solution set.