Question

Identify the pair of linear equations that have the same solution set.

2(3g+5)−2g=2(4−g) and −36h/6=2

3(m+1)=10 and 6n+3=6−n

4k−5=2k and 8j−2=3j

4=k/2+5 and 4j−14/3+5=3

Answer 1

To find the pair of linear equations that have the same solution set, we need to solve each pair of equations and compare their solution sets.

1. **First Pair of Equations:**
- \( 2(3g + 5) - 2g = 2(4 - g) \)
- Simplifying:
\[
6g + 10 - 2g = 8 - 2g
\]
\[
4g + 10 = 8 - 2g
\]
\[
4g + 2g = 8 - 10
\]
\[
6g = -2
\]
\[
g = -\frac{1}{3}
\]

- \( -\frac{36h}{6} = 2 \)
- Simplifying:
\[
-6h = 2
\]
\[
h = -\frac{1}{3}
\]

**Both equations have a solution of \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).**

2. **Second Pair of Equations:**
- \( 3(m + 1) = 10 \)
- Simplifying:
\[
3m + 3 = 10
\]
\[
3m = 7
\]
\[
m = \frac{7}{3}
\]

- \( 6n + 3 = 6 - n \)
- Simplifying:
\[
6n + n = 6 - 3
\]
\[
7n = 3
\]
\[
n = \frac{3}{7}
\]

**These do not have the same solution set.**

3. **Third Pair of Equations:**
- \( 4k - 5 = 2k \)
- Simplifying:
\[
4k - 2k = 5 + 2
\]
\[
2k = 7
\]
\[
k = \frac{7}{2}
\]

- \( 8j - 2 = 3j \)
- Simplifying:
\[
8j - 3j = 2
\]
\[
5j = 2
\]
\[
j = \frac{2}{5}
\]

**These do not have the same solution set.**

4. **Fourth Pair of Equations:**
- \( 4 = \frac{k}{2} + 5 \)
- Simplifying:
\[
\frac{k}{2} = 4 - 5
\]
\[
\frac{k}{2} = -1
\]
\[
k = -2
\]

- \( \frac{4j - 14}{3} + 5 = 3 \)
- Simplifying:
\[
\frac{4j - 14}{3} = 3 - 5
\]
\[
\frac{4j - 14}{3} = -2
\]
Multiply by 3:
\[
4j - 14 = -6
\]
\[
4j = 8
\]
\[
j = 2
\]

**These do not have the same solution set.**

### Conclusion
Upon checking all pairs, the first pair of equations \( 2(3g+5)-2g=2(4-g) \) and \( -\frac{36h}{6}=2 \) both yield \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).

Thus, the pair of linear equations that have the same solution set is:

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