To identify the pairs of linear equations that have the same solution set, we need to solve each pair of equations.
-
Equations:
- \(4k - 5 = 2k\)
- \(8j - 2 = 3j\)
Solving for \(k\): \[ 4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2} \]
Solving for \(j\): \[ 8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5} \]
The solution sets are: \(k = \frac{5}{2}\) and \(j = \frac{2}{5}\), so they do not have the same solution set.
-
Equations:
- \(29(3g + 5) - 2g = 2(4 - g)\)
- \(-36h / 6 = 2\)
Solving for \(g\): \[ 29(3g + 5) - 2g = 2(4 - g) \] Expanding and simplifying: \[ 87g + 145 - 2g = 8 - 2g \] \[ 89g + 145 = 8 \] \[ 89g = 8 - 145 \implies 89g = -137 \implies g = -\frac{137}{89} \]
Solving for \(h\): \[ -36h / 6 = 2 \implies -6h = 2 \implies h = -\frac{1}{3} \]
The solution sets are: \(g = -\frac{137}{89}\) and \(h = -\frac{1}{3}\), so they do not have the same solution set.
-
Equations:
- \(3(m + 1) = 10\)
- \(6n + 3 = 6 - n\)
Solving for \(m\): \[ 3(m + 1) = 10 \implies 3m + 3 = 10 \implies 3m = 7 \implies m = \frac{7}{3} \]
Solving for \(n\): \[ 6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7} \]
The solution sets are: \(m = \frac{7}{3}\) and \(n = \frac{3}{7}\), so they do not have the same solution set.
-
Equations:
- \(4 = \frac{k}{2} + 5\)
- \(\frac{4j - 14}{3} + 5 = 3\)
Solving for \(k\): \[ 4 = \frac{k}{2} + 5 \implies \frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2 \]
Solving for \(j\): \[ \frac{4j - 14}{3} + 5 = 3 \implies \frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2 \] \[ 4j - 14 = -6 \implies 4j = 8 \implies j = 2 \]
The solution sets are: \(k = -2\) and \(j = 2\), so they do not have the same solution set.
In conclusion, none of the given pairs of linear equations have the same solution set.
1 answer