Let's solve each equation to identify their solution sets.
Equation A: \[ \frac{4}{5}(3f - 7) = \frac{2}{5}(4f - 8) \] Multiply both sides by 5 to eliminate the fractions: \[ 4(3f - 7) = 2(4f - 8) \] Distributing both sides: \[ 12f - 28 = 8f - 16 \] Rearranging to isolate \(f\): \[ 12f - 8f = -16 + 28 \] \[ 4f = 12 \] \[ f = 3 \]
Equation B: \[ \frac{5}{9}g - 2 = \frac{7}{9} \] Add 2 to both sides: \[ \frac{5}{9}g = \frac{7}{9} + 2 \] Convert 2 to a fraction with a denominator of 9, which is \(\frac{18}{9}\): \[ \frac{5}{9}g = \frac{25}{9} \] Multiply both sides by \(\frac{9}{5}\): \[ g = 5 \]
Equation C: \[ -2(4 - 2h) = -3(1 - h) \] Distributing both sides: \[ -8 + 4h = -3 + 3h \] Rearranging to isolate \(h\): \[ 4h - 3h = -3 + 8 \] \[ h = 5 \]
Now, we summarize the solutions:
- Equation A has \(f = 3\)
- Equation B has \(g = 5\)
- Equation C has \(h = 5\)
From this analysis, the solution sets are:
- A has a solution of 3.
- B has a solution of 5.
- C has a solution of 5.
Equations B and C have the same solution (5), which means there are 2 equations that share the same solution set.
The final answer is: 2