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 find the pair of linear equations that have the same solution set, we need to simplify the equations and check if they can be transformed into the same form.
1. **3(m + 1) = 10**:
- Distributing gives: \( 3m + 3 = 10 \)
- Rearranging gives: \( 3m = 7 \) or \( m = \frac{7}{3} \)
**6n + 3 = 6 - n**:
- Rearranging gives: \( 6n + n = 6 - 3 \)
- \( 7n = 3 \) or \( n = \frac{3}{7} \)
These do **not** have the same solution set.
2. **4k - 5 = 2k**:
- Rearranging gives: \( 4k - 2k = 5 \)
- \( 2k = 5 \) or \( k = \frac{5}{2} \)
**8j - 2 = 3j**:
- Rearranging gives: \( 8j - 3j = 2 \)
- \( 5j = 2 \) or \( j = \frac{2}{5} \)
These do **not** have the same solution set.
3. **2(3g + 5) - 2g = 2(4 - g)**:
- Expanding gives: \( 6g + 10 - 2g = 8 - 2g \)
- Simplifying gives: \( 4g + 10 = 8 - 2g \)
- Rearranging gives: \( 4g + 2g = 8 - 10 \)
- \( 6g = -2 \) or \( g = -\frac{1}{3} \)
**−36h/6 = 2**:
- Simplifying gives: \( -6h = 2 \)
- Rearranging gives: \( h = -\frac{1}{3} \)
These **do** have the same solution set.
4. **4 = k^2 + 5**:
- Rearranging gives: \( k^2 = -1 \), which has no real solutions.
**4j − 143 + 5 = 3**:
- Rearranging gives: \( 4j - 138 = 3 \)
- Simplifying gives: \( 4j = 141 \) or \( j = 35.25 \)
These do **not** have the same solution set.
From the analysis, the pair of linear equations that have the same solution set is:
**2(3g + 5) - 2g = 2(4 - g) and −36h/6 = 2.**
1. **3(m + 1) = 10**:
- Distributing gives: \( 3m + 3 = 10 \)
- Rearranging gives: \( 3m = 7 \) or \( m = \frac{7}{3} \)
**6n + 3 = 6 - n**:
- Rearranging gives: \( 6n + n = 6 - 3 \)
- \( 7n = 3 \) or \( n = \frac{3}{7} \)
These do **not** have the same solution set.
2. **4k - 5 = 2k**:
- Rearranging gives: \( 4k - 2k = 5 \)
- \( 2k = 5 \) or \( k = \frac{5}{2} \)
**8j - 2 = 3j**:
- Rearranging gives: \( 8j - 3j = 2 \)
- \( 5j = 2 \) or \( j = \frac{2}{5} \)
These do **not** have the same solution set.
3. **2(3g + 5) - 2g = 2(4 - g)**:
- Expanding gives: \( 6g + 10 - 2g = 8 - 2g \)
- Simplifying gives: \( 4g + 10 = 8 - 2g \)
- Rearranging gives: \( 4g + 2g = 8 - 10 \)
- \( 6g = -2 \) or \( g = -\frac{1}{3} \)
**−36h/6 = 2**:
- Simplifying gives: \( -6h = 2 \)
- Rearranging gives: \( h = -\frac{1}{3} \)
These **do** have the same solution set.
4. **4 = k^2 + 5**:
- Rearranging gives: \( k^2 = -1 \), which has no real solutions.
**4j − 143 + 5 = 3**:
- Rearranging gives: \( 4j - 138 = 3 \)
- Simplifying gives: \( 4j = 141 \) or \( j = 35.25 \)
These do **not** have the same solution set.
From the analysis, the pair of linear equations that have the same solution set is:
**2(3g + 5) - 2g = 2(4 - g) and −36h/6 = 2.**