Question
dentify 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
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
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
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
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
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
Answers
GPT-4o mini
To determine which pair of linear equations has the same solution set, we need to simplify each pair of equations individually.
1. **Equations: \( 3(m+1) = 10 \) and \( 6n + 3 = 6 - n \)**
- For the first equation:
\[
3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
- For the second equation:
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
2. **Equations: \( 2(3g + 5) - 2g = 2(4 - g) \) and \( -36h/6 = 2 \)**
- For the first equation:
\[
2(3g + 5) - 2g = 2(4 - g) \implies 6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 - 2g
\]
\[
6g + 10 = 8 \implies 6g = -2 \implies g = -\frac{1}{3}
\]
- For the second equation:
\[
-36h/6 = 2 \implies -6h = 2 \implies h = -\frac{1}{3}
\]
3. **Equations: \( 4k - 5 = 2k \) and \( 8j - 2 = 3j \)**
- For the first equation:
\[
4k - 5 = 2k \implies 4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
- For the second equation:
\[
8j - 2 = 3j \implies 8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
4. **Equations: \( 4 = k^2 + 5 \) and \( 4j - 143 + 5 = 3 \)**
- For the first equation:
\[
4 = k^2 + 5 \implies k^2 = 4 - 5 \implies k^2 = -1 \text{ (no real solution)}
\]
- For the second equation:
\[
4j - 143 + 5 = 3 \implies 4j - 138 = 3 \implies 4j = 141 \implies j = \frac{141}{4}
\]
After analyzing the equations, we see that:
- The first pair has no common solution because \( m = \frac{7}{3} \) and \( n = \frac{3}{7} \).
- The second pair has the same solution \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).
- The third pair is also inconsistent; \( k = \frac{5}{2} \) and \( j = \frac{2}{5} \).
- The fourth pair has no real solutions.
Thus, the pair of linear equations that have the same solution set is:
**2(3g+5)−2g=2(4−g) and −36h/6=2**.
1. **Equations: \( 3(m+1) = 10 \) and \( 6n + 3 = 6 - n \)**
- For the first equation:
\[
3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]
- For the second equation:
\[
6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
2. **Equations: \( 2(3g + 5) - 2g = 2(4 - g) \) and \( -36h/6 = 2 \)**
- For the first equation:
\[
2(3g + 5) - 2g = 2(4 - g) \implies 6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 - 2g
\]
\[
6g + 10 = 8 \implies 6g = -2 \implies g = -\frac{1}{3}
\]
- For the second equation:
\[
-36h/6 = 2 \implies -6h = 2 \implies h = -\frac{1}{3}
\]
3. **Equations: \( 4k - 5 = 2k \) and \( 8j - 2 = 3j \)**
- For the first equation:
\[
4k - 5 = 2k \implies 4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]
- For the second equation:
\[
8j - 2 = 3j \implies 8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
4. **Equations: \( 4 = k^2 + 5 \) and \( 4j - 143 + 5 = 3 \)**
- For the first equation:
\[
4 = k^2 + 5 \implies k^2 = 4 - 5 \implies k^2 = -1 \text{ (no real solution)}
\]
- For the second equation:
\[
4j - 143 + 5 = 3 \implies 4j - 138 = 3 \implies 4j = 141 \implies j = \frac{141}{4}
\]
After analyzing the equations, we see that:
- The first pair has no common solution because \( m = \frac{7}{3} \) and \( n = \frac{3}{7} \).
- The second pair has the same solution \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).
- The third pair is also inconsistent; \( k = \frac{5}{2} \) and \( j = \frac{2}{5} \).
- The fourth pair has no real solutions.
Thus, the pair of linear equations that have the same solution set is:
**2(3g+5)−2g=2(4−g) and −36h/6=2**.