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.
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First Pair of Equations:
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\( 2(3g + 5) - 2g = 2(4 - g) \)
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Simplifying: \[ 6g + 10 - 2g = 8 - 2g \] \[ 4g + 10 = 8 - 2g \] \[ 4g + 2g = 8 - 10 \] \[ 6g = -2 \] \[ g = -\frac{1}{3} \]
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\( -\frac{36h}{6} = 2 \)
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Simplifying: \[ -6h = 2 \] \[ h = -\frac{1}{3} \]
Both equations have a solution of \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).
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Second Pair of Equations:
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\( 3(m + 1) = 10 \)
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Simplifying: \[ 3m + 3 = 10 \] \[ 3m = 7 \] \[ m = \frac{7}{3} \]
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\( 6n + 3 = 6 - n \)
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Simplifying: \[ 6n + n = 6 - 3 \] \[ 7n = 3 \] \[ n = \frac{3}{7} \]
These do not have the same solution set.
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Third Pair of Equations:
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\( 4k - 5 = 2k \)
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Simplifying: \[ 4k - 2k = 5 + 2 \] \[ 2k = 7 \] \[ k = \frac{7}{2} \]
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\( 8j - 2 = 3j \)
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Simplifying: \[ 8j - 3j = 2 \] \[ 5j = 2 \] \[ j = \frac{2}{5} \]
These do not have the same solution set.
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Fourth Pair of Equations:
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\( 4 = \frac{k}{2} + 5 \)
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Simplifying: \[ \frac{k}{2} = 4 - 5 \] \[ \frac{k}{2} = -1 \] \[ k = -2 \]
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\( \frac{4j - 14}{3} + 5 = 3 \)
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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.
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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 \( -\frac{36h}{6}=2 \).