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

Question

1. 2(3g+5)-2g=2(4-g) and -36h/6=2

2. 4k-5=2k and 8j-2=3j

3. 3(m+1)=10 and 6n+3=6-n

4. 4= k/2 + 5 and 4j-14/3+5=3

Answer 1

To determine which pair of linear equations has the same solution set, let's solve each pair of equations:

1. First Pair: \[ 2(3g + 5) - 2g = 2(4 - g) \] Expanding and simplifying: \[ 6g + 10 - 2g = 8 - 2g \implies 4g + 10 = 8 - 2g \implies 6g = -2 \implies g = -\frac{1}{3} \]

   The second equation is: \[ -\frac{36h}{6} = 2 \implies -6h = 2 \implies h = -\frac{1}{3} \]

   Both variables yield \( -\frac{1}{3} \), but this does not mean both equations have the same solution set, as they are independent equations.

2. Second Pair: \[ 4k - 5 = 2k \implies 2k = 5 \implies k = \frac{5}{2} \]

   The second equation: \[ 8j - 2 = 3j \implies 5j = 2 \implies j = \frac{2}{5} \]

   These have different solutions (\(k \neq j\)).

3. Third Pair: \[ 3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3} \]

   The second equation: \[ 6n + 3 = 6 - n \implies 7n = 3 \implies n = \frac{3}{7} \]

   Different solutions (\(m \neq n\)).

4. Fourth Pair: \[ 4 = \frac{k}{2} + 5 \implies -1 = \frac{k}{2} \implies k = -2 \]

   The second equation: \[ \frac{4j - 14}{3} + 5 = 3 \implies \frac{4j - 14}{3} = -2 \implies 4j - 14 = -6 \implies 4j = 8 \implies j = 2 \]

   Again, different solutions (\(k \neq j\)).

Conclusion: Each pair resulted in different values for their respective variables. Therefore, none of the pairs have the same solution set. However, if we are strictly identifying pairs based on the question format, Pair 1 does yield \(g = h\), even if they are different equations.

Further review shows none match \(m = j\) or similar.

Thus, if we are looking for a definitive answer, consider that no pairs yield the same solution set directly, leaving us to infer Pair 1 or to reconvene on strict comparisons of all statements initially until an option might distinctly show it.

Let me know if further exploration or specifics on resolution paths or variation checks are necessary!

Answer 2

1 2 3 or 4?

Answer 3

The pair of linear equations that have the same solution set is none of the above. However, upon reevaluating each option with a focus on which seems closest:

1. Option 1: \( g = -\frac{1}{3} \), \( h = -\frac{1}{3} \) (same solution)
2. Option 2: \( k = \frac{5}{2} \), \( j = \frac{2}{5} \) (different)
3. Option 3: \( m = \frac{7}{3} \), \( n = \frac{3}{7} \) (different)
4. Option 4: \( k = -2 \), \( j = 2 \) (different)

Correct Answer: 1. The first pair of equations has the same solution, which is \( -\frac{1}{3} \) for both \( g \) and \( h \).