Question
Which of the linear equations has the same solution set as 9d4+3−2d=−2(3+d) ?(1 point)
Responses
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
3x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
35−15b=95
Responses
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
3x5−10=−7
Start Fraction 3 x over 5 End Fraction minus 10 equals negative 7
35−15b=95
Answers
Answer
at least an answer similar
Answered by
GPT-4o mini
To determine which of the given linear equations has the same solution set as \( 9d^4 + 3 - 2d = -2(3 + d) \), we first need to simplify the equation.
1. Start with the original equation:
\[
9d^4 + 3 - 2d = -2(3 + d)
\]
2. Expand the right side:
\[
9d^4 + 3 - 2d = -6 - 2d
\]
3. Now, add \( 2d \) to both sides:
\[
9d^4 + 3 = -6
\]
4. Then, add \( 6 \) to both sides:
\[
9d^4 + 9 = 0
\]
5. Finally, subtract \( 9 \) from both sides to isolate \( d^4 \):
\[
9d^4 = -9
\]
\[
d^4 = -1
\]
Since \( d^4 = -1 \) has no real solutions (the fourth power of any real number cannot result in a negative value), the original equation has no solutions in the real numbers.
Now, let's analyze the given options to find which one has the same solution set (which would also be no solutions):
1. **Option 1**: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)
This can be simplified as follows:
\[
14 = \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x
\]
Combine like terms:
\[
14 = \frac{15}{4}x - \frac{19}{4}
\]
Multiply everything by 4 to eliminate the fraction:
\[
56 = 15x - 19
\]
\[
75 = 15x
\]
\[
x = 5
\]
This equation has a solution.
2. **Option 2**: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)
Simplifying gives:
\[
4w - (14 - 6w) = \frac{w}{3} - 6
\]
\[
4w + 6w - 14 = \frac{w}{3} - 6
\]
\[
10w - 14 = \frac{w}{3} - 6
\]
Multiply everything by 3:
\[
30w - 42 = w - 18
\]
\[
29w = 24
\]
\[
w = \frac{24}{29}
\]
This equation has a solution.
3. **Option 3**: \( \frac{3x}{5} - 10 = -7 \)
Simplifying gives:
\[
\frac{3x}{5} = 3
\]
\[
3x = 15
\]
\[
x = 5
\]
This equation has a solution.
4. **Option 4**: \( 35 - 15b = 95 \)
Rearranging gives:
\[
-15b = 95 - 35
\]
\[
-15b = 60
\]
\[
b = -4
\]
This equation has a solution.
Since the original equation has no solution, we are looking for an equation from the options that also has no solution.
None of the options above yield no solutions. Therefore, none of these equations will have the same solution set (which has no solutions) as the original equation:
\[
9d^4 + 3 - 2d = -2(3 + d)
\]
1. Start with the original equation:
\[
9d^4 + 3 - 2d = -2(3 + d)
\]
2. Expand the right side:
\[
9d^4 + 3 - 2d = -6 - 2d
\]
3. Now, add \( 2d \) to both sides:
\[
9d^4 + 3 = -6
\]
4. Then, add \( 6 \) to both sides:
\[
9d^4 + 9 = 0
\]
5. Finally, subtract \( 9 \) from both sides to isolate \( d^4 \):
\[
9d^4 = -9
\]
\[
d^4 = -1
\]
Since \( d^4 = -1 \) has no real solutions (the fourth power of any real number cannot result in a negative value), the original equation has no solutions in the real numbers.
Now, let's analyze the given options to find which one has the same solution set (which would also be no solutions):
1. **Option 1**: \( 14 = \frac{7}{4}x - \frac{19}{4} + 2x \)
This can be simplified as follows:
\[
14 = \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x
\]
Combine like terms:
\[
14 = \frac{15}{4}x - \frac{19}{4}
\]
Multiply everything by 4 to eliminate the fraction:
\[
56 = 15x - 19
\]
\[
75 = 15x
\]
\[
x = 5
\]
This equation has a solution.
2. **Option 2**: \( 4w - 2(7 - 3w) = \frac{w}{3} - 6 \)
Simplifying gives:
\[
4w - (14 - 6w) = \frac{w}{3} - 6
\]
\[
4w + 6w - 14 = \frac{w}{3} - 6
\]
\[
10w - 14 = \frac{w}{3} - 6
\]
Multiply everything by 3:
\[
30w - 42 = w - 18
\]
\[
29w = 24
\]
\[
w = \frac{24}{29}
\]
This equation has a solution.
3. **Option 3**: \( \frac{3x}{5} - 10 = -7 \)
Simplifying gives:
\[
\frac{3x}{5} = 3
\]
\[
3x = 15
\]
\[
x = 5
\]
This equation has a solution.
4. **Option 4**: \( 35 - 15b = 95 \)
Rearranging gives:
\[
-15b = 95 - 35
\]
\[
-15b = 60
\]
\[
b = -4
\]
This equation has a solution.
Since the original equation has no solution, we are looking for an equation from the options that also has no solution.
None of the options above yield no solutions. Therefore, none of these equations will have the same solution set (which has no solutions) as the original equation:
\[
9d^4 + 3 - 2d = -2(3 + d)
\]
Answered by
GPT-4o mini
I apologize for the confusion in the previous response. To clarify, we need to determine which of the provided equations does **not** have any solutions, as the given equation produces no real solutions.
Let’s reconsider each option briefly:
1. **Option 1**:
\[
14 = \frac{7}{4}x - \frac{19}{4} + 2x
\]
This equation can be simplified, and it produces a real solution.
2. **Option 2**:
\[
4w - 2(7 - 3w) = \frac{w}{3} - 6
\]
This equation can also be simplified, and it produces a real solution.
3. **Option 3**:
\[
\frac{3x}{5} - 10 = -7
\]
Solving this yields a real solution.
4. **Option 4**:
\[
35 - 15b = 95
\]
Rearranging produces a real solution.
To determine if any of the tracks lead to contradictions (thus yielding no solutions), we can look specifically for cases that might lead to a false statement, such as yields \(0 = c\), where \(c\) is a nonzero constant.
None of the options provided yield a contradiction, and all options will yield solutions. Therefore, none of the equations have the same solution set as the original equation \(9d^4 + 3 - 2d = -2(3 + d)\).
Since your request asks for an answer that is similar, if I had to select the best response while acknowledging that all provided equations yield valid solutions, I’d suggest that none match your original equation since they all produce some solutions, albeit none leading to no solution.
If you are looking for a specific equation similar to the original (which has no solutions), one you might write could be of the form:
\[
k \cdot d = k \cdot d + c \quad (c \neq 0)
\]
This format will always yield a contradiction \(0 = c\).
If you need help with anything more specific or need a confirmation of certain choices, please let me know!
Let’s reconsider each option briefly:
1. **Option 1**:
\[
14 = \frac{7}{4}x - \frac{19}{4} + 2x
\]
This equation can be simplified, and it produces a real solution.
2. **Option 2**:
\[
4w - 2(7 - 3w) = \frac{w}{3} - 6
\]
This equation can also be simplified, and it produces a real solution.
3. **Option 3**:
\[
\frac{3x}{5} - 10 = -7
\]
Solving this yields a real solution.
4. **Option 4**:
\[
35 - 15b = 95
\]
Rearranging produces a real solution.
To determine if any of the tracks lead to contradictions (thus yielding no solutions), we can look specifically for cases that might lead to a false statement, such as yields \(0 = c\), where \(c\) is a nonzero constant.
None of the options provided yield a contradiction, and all options will yield solutions. Therefore, none of the equations have the same solution set as the original equation \(9d^4 + 3 - 2d = -2(3 + d)\).
Since your request asks for an answer that is similar, if I had to select the best response while acknowledging that all provided equations yield valid solutions, I’d suggest that none match your original equation since they all produce some solutions, albeit none leading to no solution.
If you are looking for a specific equation similar to the original (which has no solutions), one you might write could be of the form:
\[
k \cdot d = k \cdot d + c \quad (c \neq 0)
\]
This format will always yield a contradiction \(0 = c\).
If you need help with anything more specific or need a confirmation of certain choices, please let me know!
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