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
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
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
4=k2+5 and 4j−143+5=3
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
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
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
4=k2+5 and 4j−143+5=3
Answers
GPT-4o mini
To identify the pair of linear equations that have the same solution set, we have to solve each pair and see if they yield the same values for their respective variables.
1. **For the first pair:**
- Equation 1: \( 3(m + 1) = 10 \)
- Equation 2: \( 6n + 3 = 6 - n \)
**Solving Equation 1:**
\[
3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
**Solving Equation 2:**
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
**These equations do not have the same solution.**
2. **For the second pair:**
- Equation 1: \( 4k - 5 = 2k \)
- Equation 2: \( 8j - 2 = 3j \)
**Solving Equation 1:**
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
**Solving Equation 2:**
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
**These equations do not have the same solution.**
3. **For the third pair:**
- Equation 1: \( 2(3g + 5) - 2g = 2(4 - g) \)
- Equation 2: \( -36h / 6 = 2 \)
**Solving Equation 1:**
\[
2(3g + 5) - 2g = 2(4 - g)
\]
\[
6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 + 2g \implies 4g - 2g = 8 - 10
\implies 2g = -2 \implies g = -1
\]
**Solving Equation 2:**
\[
-36h / 6 = 2 \implies -6h = 2 \implies h = -\frac{1}{3}
\]
**These equations do not have the same solution.**
4. **For the fourth pair:**
- Equation 1: \( 4 = k^2 + 5 \)
- Equation 2: \( 4j - 14 / 3 + 5 = 3 \)
**Solving Equation 1:**
\[
k^2 = 4 - 5 \implies k^2 = -1 \implies k \text{ has no real solution.}
\]
**Solving Equation 2:**
\[
4j - \frac{14}{3} + 5 = 3 \implies 4j - \frac{14}{3} + \frac{15}{3} = 3
\]
\[
4j + \frac{1}{3} = 3 \implies 4j = 3 - \frac{1}{3} = \frac{9}{3} - \frac{1}{3} = \frac{8}{3}
\]
\[
j = \frac{2}{3}
\]
**These equations do not have the same solution.**
After analyzing all pairs of equations, none of them share the same solution set. Therefore, it appears that none of the provided pairs of equations have the same solution set.
1. **For the first pair:**
- Equation 1: \( 3(m + 1) = 10 \)
- Equation 2: \( 6n + 3 = 6 - n \)
**Solving Equation 1:**
\[
3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
**Solving Equation 2:**
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
**These equations do not have the same solution.**
2. **For the second pair:**
- Equation 1: \( 4k - 5 = 2k \)
- Equation 2: \( 8j - 2 = 3j \)
**Solving Equation 1:**
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
**Solving Equation 2:**
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
**These equations do not have the same solution.**
3. **For the third pair:**
- Equation 1: \( 2(3g + 5) - 2g = 2(4 - g) \)
- Equation 2: \( -36h / 6 = 2 \)
**Solving Equation 1:**
\[
2(3g + 5) - 2g = 2(4 - g)
\]
\[
6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 + 2g \implies 4g - 2g = 8 - 10
\implies 2g = -2 \implies g = -1
\]
**Solving Equation 2:**
\[
-36h / 6 = 2 \implies -6h = 2 \implies h = -\frac{1}{3}
\]
**These equations do not have the same solution.**
4. **For the fourth pair:**
- Equation 1: \( 4 = k^2 + 5 \)
- Equation 2: \( 4j - 14 / 3 + 5 = 3 \)
**Solving Equation 1:**
\[
k^2 = 4 - 5 \implies k^2 = -1 \implies k \text{ has no real solution.}
\]
**Solving Equation 2:**
\[
4j - \frac{14}{3} + 5 = 3 \implies 4j - \frac{14}{3} + \frac{15}{3} = 3
\]
\[
4j + \frac{1}{3} = 3 \implies 4j = 3 - \frac{1}{3} = \frac{9}{3} - \frac{1}{3} = \frac{8}{3}
\]
\[
j = \frac{2}{3}
\]
**These equations do not have the same solution.**
After analyzing all pairs of equations, none of them share the same solution set. Therefore, it appears that none of the provided pairs of equations have the same solution set.