Question

Identify the pair of linear equations that have the same solution set. (1 point)
$3(m+1)=10$ and $6n+3=6-n$
$4=\frac {k}{2}+5$ and $\frac {4j-14}{3}+5=3$
$2(3g+5)-2g=2(4-g)$ and $\frac {-36h}{6}=2$
$4k-5=2kand8j-2=3j$

Answer 1

To determine which pair of linear equations has the same solution set, we need to solve each pair of equations and check if they yield equivalent solutions.

1. **Equations:**
\[
3(m + 1) = 10
\]
Simplifying:
\[
3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3}
\]

Second equation:
\[
6n + 3 = 6 - n
\]
Rearranging:
\[
6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}
\]
(These equations are not equivalent in solutions.)

2. **Equations:**
\[
4 = \frac{k}{2} + 5
\]
Simplifying:
\[
\frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2
\]

Second equation:
\[
\frac{4j - 14}{3} + 5 = 3
\]
Rearranging:
\[
\frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2
\]
Multiplying through by 3:
\[
4j - 14 = -6 \implies 4j = 8 \implies j = 2
\]
(These equations are not equivalent in solutions.)

3. **Equations:**
\[
2(3g + 5) - 2g = 2(4 - g)
\]
Expanding:
\[
6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 - 2g
\]
Rearranging:
\[
4g + 2g = 8 - 10 \implies 6g = -2 \implies g = -\frac{1}{3}
\]

Second equation:
\[
\frac{-36h}{6} = 2
\]
Simplifying:
\[
-6h = 2 \implies h = -\frac{1}{3}
\]
(These equations are equivalent in solutions.)

4. **Equations:**
\[
4k - 5 = 2k
\]
Rearranging:
\[
4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}
\]

Second equation:
\[
8j - 2 = 3j
\]
Rearranging:
\[
8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}
\]
(These equations are not equivalent in solutions.)

**Conclusion:**
The only pair of equations that have the same solution set is:
\[
2(3g + 5) - 2g = 2(4 - g) \quad \text{and} \quad \frac{-36h}{6} = 2
\]
So, the final answer is:
\[
\boxed{(2(3g+5)-2g=2(4-g) \text{ and } \frac {-36h}{6}=2)}
\]