Question

Identify the pair of linear equations that have the same solution set.(1 point)
Responses

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

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

3(m+1)=10
and 6n+3=6−n

Answer 1

To identify the pair of linear equations that have the same solution set, we can simplify each pair of equations and check if they yield the same solution.

1. **First Pair:**
- Equation 1: \( 4 = k^2 + 5 \) → \( k^2 = 4 - 5 \) → \( k^2 = -1 \) (no real solution)
- Equation 2: \( 4j - \frac{14}{3} + 5 = 3 \) leads to \( 4j - \frac{14}{3} = -2 \) → \( 4j = -2 + \frac{14}{3} \)
- Solution comparison not necessary since first has no real solution.

2. **Second Pair:**
- First equation: \( 2(3g + 5) - 2g = 2(4 - g) \)
- Distribute: \( 6g + 10 - 2g = 8 - 2g \)
- Combine: \( 4g + 10 = 8 - 2g \)
- Rearranging gives: \( 6g = -2 \) → \( g = -\frac{1}{3} \)
- Second equation: \(-\frac{36h}{6} = 2\) → \( -6h = 2 \) → \( h = -\frac{1}{3} \)
- **Same solution:** \( g = h = -\frac{1}{3} \)

3. **Third Pair:**
- First equation: \( 4k - 5 = 2k \)
- Combine gives: \( 2k = 5 \) → \( k = \frac{5}{2} \)
- Second equation: \( 8j - 2 = 3j \)
- Rearranging gives: \( 5j = 2 \) → \( j = \frac{2}{5} \)
- **Different solutions.**

4. **Fourth Pair:**
- First equation: \( 3(m + 1) = 10 \)
- Solving gives: \( m + 1 = \frac{10}{3} \) → \( m = \frac{7}{3} \)
- Second equation: \( 6n + 3 = 6 - n \)
- Rearranging: \( 6n + n = 3 - 6 \) → \( 7n = -3 \) → \( n = -\frac{3}{7} \)
- **Different solutions.**

**Conclusion:** The pair of linear equations that have the same solution set is:

**2(3g + 5) - 2g = 2(4 - g)** and **\(-\frac{36h}{6} = 2\)**.