Question
Identify the pair of linear equations that have the same solution set.(1 point)
Responses
4k−5=2k
and 8j−2=3j
4 k minus 5 equals 2 k and 8 j minus 2 equals 3 j
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
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
2(3g+5)−2g=2(4−g)
and −36h6=2
2 left parenthesis 3 g plus 5 right parenthesis minus 2 g equals 2 left parenthesis 4 minus g right parenthesis and Start Fraction negative 36 h over 6 End Fraction equals 2
Responses
4k−5=2k
and 8j−2=3j
4 k minus 5 equals 2 k and 8 j minus 2 equals 3 j
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
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
2(3g+5)−2g=2(4−g)
and −36h6=2
2 left parenthesis 3 g plus 5 right parenthesis minus 2 g equals 2 left parenthesis 4 minus g right parenthesis and Start Fraction negative 36 h over 6 End Fraction equals 2
Answers
GPT-4o mini
To determine which pair of linear equations has the same solution set, we'll solve each pair step by step.
1. **For the first pair:**
\[
4k - 5 = 2k
\]
Rearranging gives:
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
The solution is \( k = \frac{5}{2} \).
\[
8j - 2 = 3j
\]
Rearranging gives:
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
The solutions are not the same.
2. **For the second pair:**
\[
4 = \frac{k}{2} + 5
\]
Rearranging gives:
\[
\frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2
\]
The solution is \( k = -2 \).
\[
\frac{4j - 14}{3} + 5 = 3
\]
Rearranging gives:
\[
\frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2
\]
Multiplying both sides by 3:
\[
4j - 14 = -6 \implies 4j = 14 - 6 \implies 4j = 8 \implies j = 2
\]
The solutions are not the same.
3. **For the third pair:**
\[
3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
The solution is \( m = \frac{7}{3} \).
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
The solutions are not the same.
4. **For the fourth pair:**
\[
2(3g + 5) - 2g = 2(4 - g)
\]
Expanding gives:
\[
6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 \implies 4g = -2 \implies g = -\frac{1}{2}
\]
The solution is \( g = -\frac{1}{2} \).
\[
-\frac{36h}{6} = 2
\]
Simplifying gives:
\[
-6h = 2 \implies h = -\frac{1}{3}
\]
The solutions are not the same.
**Conclusion:** Comparing the pairs, none of the pairs of equations share the same solution set. Please check for any errors or miscommunication in the equations provided.
1. **For the first pair:**
\[
4k - 5 = 2k
\]
Rearranging gives:
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
The solution is \( k = \frac{5}{2} \).
\[
8j - 2 = 3j
\]
Rearranging gives:
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
The solutions are not the same.
2. **For the second pair:**
\[
4 = \frac{k}{2} + 5
\]
Rearranging gives:
\[
\frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2
\]
The solution is \( k = -2 \).
\[
\frac{4j - 14}{3} + 5 = 3
\]
Rearranging gives:
\[
\frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2
\]
Multiplying both sides by 3:
\[
4j - 14 = -6 \implies 4j = 14 - 6 \implies 4j = 8 \implies j = 2
\]
The solutions are not the same.
3. **For the third pair:**
\[
3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
The solution is \( m = \frac{7}{3} \).
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
The solutions are not the same.
4. **For the fourth pair:**
\[
2(3g + 5) - 2g = 2(4 - g)
\]
Expanding gives:
\[
6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 \implies 4g = -2 \implies g = -\frac{1}{2}
\]
The solution is \( g = -\frac{1}{2} \).
\[
-\frac{36h}{6} = 2
\]
Simplifying gives:
\[
-6h = 2 \implies h = -\frac{1}{3}
\]
The solutions are not the same.
**Conclusion:** Comparing the pairs, none of the pairs of equations share the same solution set. Please check for any errors or miscommunication in the equations provided.