Question
Identify the pair of linear equations that have the same solution set.(1 point)
Responses
3(m+1)=10 and 6n+3=6−n
3 left parenthesis m plus 1 right parenthesis equals 10 and 6 n plus 3 equals 6 minus n
4=k2+5 and 4j−143+5=3
4 equals Start Fraction k over 2 End Fraction plus 5 and Start Fraction 4 j minus 14 over 3 End Fraction plus 5 equals 3
4k−5=2k and 8j−2=3j
4 k minus 5 equals 2 k and 8 j minus 2 equals 3 j
2(3g+5)−2g=2(4−g) and −36h6=2
Responses
3(m+1)=10 and 6n+3=6−n
3 left parenthesis m plus 1 right parenthesis equals 10 and 6 n plus 3 equals 6 minus n
4=k2+5 and 4j−143+5=3
4 equals Start Fraction k over 2 End Fraction plus 5 and Start Fraction 4 j minus 14 over 3 End Fraction plus 5 equals 3
4k−5=2k and 8j−2=3j
4 k minus 5 equals 2 k and 8 j minus 2 equals 3 j
2(3g+5)−2g=2(4−g) and −36h6=2
Answers
GPT-4o mini
To determine the pair of linear equations that have the same solution set, we can simplify each equation individually and check if they yield the same linear equation.
1. **3(m + 1) = 10 and 6n + 3 = 6 − n**
- For the first equation:
\[
3(m + 1) = 10
\]
simplifies to:
\[
3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
- For the second equation:
\[
6n + 3 = 6 - n
\]
simplifies to:
\[
6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
Since they yield different solutions, this pair does not have the same solution set.
2. **4 = k/2 + 5 and (4j - 14)/3 + 5 = 3**
- For the first equation:
\[
4 = \frac{k}{2} + 5
\]
simplifies to:
\[
\frac{k}{2} = -1 \implies k = -2
\]
- For the second equation:
\[
\frac{4j - 14}{3} + 5 = 3
\]
simplifies to:
\[
\frac{4j - 14}{3} = -2 \implies 4j - 14 = -6 \implies 4j = 8 \implies j = 2
\]
Since they yield different solutions, this pair does not have the same solution set.
3. **4k − 5 = 2k and 8j − 2 = 3j**
- For the first equation:
\[
4k - 5 = 2k
\]
simplifies to:
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
- For the second equation:
\[
8j - 2 = 3j
\]
simplifies to:
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
Since they yield different solutions, this pair does not have the same solution set.
4. **2(3g + 5) - 2g = 2(4 - g) and -36h/6 = 2**
- For the first equation:
\[
2(3g + 5) - 2g = 2(4 - g)
\]
simplifies to:
\[
6g + 10 - 2g = 8 - 2g \implies 6g + 10 = 8 \implies 4g = -2 \implies g = -\frac{1}{2}
\]
- For the second equation:
\[
-\frac{36h}{6} = 2
\]
simplifies to:
\[
-6h = 2 \implies h = -\frac{1}{3}
\]
Since they yield different solutions, this pair does not have the same solution set.
After analyzing each pair of equations, none of them have the same solution set. Therefore, there appears to be an error in the provided options or assumptions, as none yield equivalent results under simplification.
If you clarify any constraints or specific focus of the question, I can reassess for potential equations that might achieve similarity. Otherwise, based on a straightforward approach, none of these pairs share the same solution in their transformations.
1. **3(m + 1) = 10 and 6n + 3 = 6 − n**
- For the first equation:
\[
3(m + 1) = 10
\]
simplifies to:
\[
3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
- For the second equation:
\[
6n + 3 = 6 - n
\]
simplifies to:
\[
6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
Since they yield different solutions, this pair does not have the same solution set.
2. **4 = k/2 + 5 and (4j - 14)/3 + 5 = 3**
- For the first equation:
\[
4 = \frac{k}{2} + 5
\]
simplifies to:
\[
\frac{k}{2} = -1 \implies k = -2
\]
- For the second equation:
\[
\frac{4j - 14}{3} + 5 = 3
\]
simplifies to:
\[
\frac{4j - 14}{3} = -2 \implies 4j - 14 = -6 \implies 4j = 8 \implies j = 2
\]
Since they yield different solutions, this pair does not have the same solution set.
3. **4k − 5 = 2k and 8j − 2 = 3j**
- For the first equation:
\[
4k - 5 = 2k
\]
simplifies to:
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
- For the second equation:
\[
8j - 2 = 3j
\]
simplifies to:
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
Since they yield different solutions, this pair does not have the same solution set.
4. **2(3g + 5) - 2g = 2(4 - g) and -36h/6 = 2**
- For the first equation:
\[
2(3g + 5) - 2g = 2(4 - g)
\]
simplifies to:
\[
6g + 10 - 2g = 8 - 2g \implies 6g + 10 = 8 \implies 4g = -2 \implies g = -\frac{1}{2}
\]
- For the second equation:
\[
-\frac{36h}{6} = 2
\]
simplifies to:
\[
-6h = 2 \implies h = -\frac{1}{3}
\]
Since they yield different solutions, this pair does not have the same solution set.
After analyzing each pair of equations, none of them have the same solution set. Therefore, there appears to be an error in the provided options or assumptions, as none yield equivalent results under simplification.
If you clarify any constraints or specific focus of the question, I can reassess for potential equations that might achieve similarity. Otherwise, based on a straightforward approach, none of these pairs share the same solution in their transformations.