Question

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

4=k2+5 and 4j−143+5=3
4 equals Start Fraction k over 2 End Fraction plus 5 and Start Fraction 4 j minus 14 over 3 End Fraction plus 5 equals 3

3(m+1)=10 and 6n+3=6−n
3 left parenthesis m plus 1 right parenthesis equals 10 and 6 n plus 3 equals 6 minus n

2(3g+5)−2g=2(4−g) and −36h6=2
2 left parenthesis 3 g plus 5 right parenthesis minus 2 g equals 2 left parenthesis 4 minus g right parenthesis and Start Fraction negative 36 h over 6 End Fraction equals 2

4k−5=2k and 8j−2=3j
4 k minus 5 equals 2 k and 8 j minus 2 equals 3 j

Answer 1

To identify which pair of linear equations have the same solution set, we can solve each pair and see if they yield the same solution for their respective variables.

1. **First Pair**:
- \( 4 = \frac{k}{2} + 5 \)
- \( 4j - \frac{14}{3} + 5 = 3 \)

For the first equation:
\[
4 - 5 = \frac{k}{2} \\
-1 = \frac{k}{2} \\
k = -2
\]

For the second equation:
\[
4j - \frac{14}{3} + 5 = 3 \\
4j + \frac{1}{3} = 3 \\
4j = 3 - \frac{1}{3} \\
4j = \frac{9}{3} - \frac{1}{3} = \frac{8}{3} \\
j = \frac{2}{3}
\]

So, this pair does not have the same solution set.

2. **Second Pair**:
- \( 3(m + 1) = 10 \)
- \( 6n + 3 = 6 - n \)

For the first equation:
\[
3m + 3 = 10 \\
3m = 7 \\
m = \frac{7}{3}
\]

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

So, this pair does not have the same solution set.

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

For the first equation:
\[
6g + 10 - 2g = 8 - 2g \\
6g + 2g + 2g = 8 - 10 \\
6g = -2 \\
g = -\frac{1}{3}
\]

For the second equation:
\[
-6h = 2 \\
h = -\frac{1}{3}
\]

This pair has the same solution set \( g = -\frac{1}{3} \) and \( h = -\frac{1}{3} \).

4. **Fourth Pair**:
- \( 4k - 5 = 2k \)
- \( 8j - 2 = 3j \)

For the first equation:
\[
4k - 2k = 5 \\
2k = 5 \\
k = \frac{5}{2}
\]

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

So, this pair does not have the same solution set.

Thus, the pair of linear equations that have the same solution set is:
**2(3g + 5) - 2g = 2(4 - g) and -36h/6 = 2**.