To find out how many of the given equations have the same solution set, we need to solve each equation and compare their solutions.
Equation A: \[ \frac{4}{5}(3f - 7) = \frac{2}{5}(4f - 8) \] First, let's eliminate the fractions by multiplying both sides by 5: \[ 4(3f - 7) = 2(4f - 8) \] Expanding both sides: \[ 12f - 28 = 8f - 16 \] Rearranging gives: \[ 12f - 8f = -16 + 28 \] \[ 4f = 12 \] \[ f = 3 \]
Equation B: \[ \frac{5}{9}g - 2 = \frac{7}{9} \] Adding 2 to both sides: \[ \frac{5}{9}g = \frac{7}{9} + 2 \] To add 2, first convert it to a fraction: \[ 2 = \frac{18}{9} \] So, \[ \frac{5}{9}g = \frac{7}{9} + \frac{18}{9} = \frac{25}{9} \] Multiplying both sides by \(\frac{9}{5}\): \[ g = \frac{9}{5} \cdot \frac{25}{9} = 5 \]
Equation C: \[ -2(4 - 2h) = -3(1 - h) \] Expanding: \[ -8 + 4h = -3 + 3h \] Rearranging gives: \[ 4h - 3h = -3 + 8 \] \[ h = 5 \]
Now, we summarize the solutions:
- Equation A has the solution \(f = 3\).
- Equation B has the solution \(g = 5\).
- Equation C has the solution \(h = 5\).
Since:
- Equation A has a solution of \(3\),
- Equation B and C both have a solution of \(5\).
Equations B and C have the same solution, but Equation A does not match with the others.
Thus, the number of equations that have the same solution set is 2 (Equations B and C).
The correct response is: 2.
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