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

1 answer

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.