To determine which pair of linear equations have the same solution set, we need to simplify each pair and check if they lead to the same variable equation.
A)
- \( 4k - 5 = 2k \)
- Rearranging gives: \[ 4k - 2k = 5 \Rightarrow 2k = 5 \Rightarrow k = \frac{5}{2} \]
- \( 8j - 2 = 3j \)
- Rearranging gives: \[ 8j - 3j = 2 \Rightarrow 5j = 2 \Rightarrow j = \frac{2}{5} \]
Not the same solution set.
B)
- \( 4 = \frac{k}{2} + 5 \)
- Rearranging gives: \[ \frac{k}{2} = 4 - 5 \Rightarrow \frac{k}{2} = -1 \Rightarrow k = -2 \]
- \( 4j - \frac{14}{3} + 5 = 3 \)
- Rearranging gives: \[ 4j - \frac{14}{3} = 3 - 5 \Rightarrow 4j - \frac{14}{3} = -2 \]
- \( 4j = -2 + \frac{14}{3} \)
- Convert -2 into thirds: \( -2 = -\frac{6}{3} \)
- Thus: \[ 4j = -\frac{6}{3} + \frac{14}{3} = \frac{8}{3} \Rightarrow j = \frac{2}{3} \]
Not the same solution set.
C)
- \( 3(m + 1) = 10 \)
- Rearranging gives: \[ 3m + 3 = 10 \Rightarrow 3m = 10 - 3 \Rightarrow 3m = 7 \Rightarrow m = \frac{7}{3} \]
- \( 6n + 3 = 6 - n \)
- Rearranging gives: \[ 6n + n = 6 - 3 \Rightarrow 7n = 3 \Rightarrow n = \frac{3}{7} \]
Not the same solution set.
D)
- \( 2(3g + 5) - 2g = 2(4 - g) \)
- Expand: \[ 6g + 10 - 2g = 8 - 2g \]
- Combine like terms: \[ 4g + 10 = 8 - 2g \]
- Rearranging gives: \[ 4g + 2g = 8 - 10 \Rightarrow 6g = -2 \Rightarrow g = -\frac{1}{3} \]
- \( -\frac{36h}{6} = 2 \)
- Simplifying gives: \[ -6h = 2 \Rightarrow h = -\frac{1}{3} \]
Both have the solution \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).
The same solution set is present in Option D.
Answer: D)
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