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Identify the number of equations that have the same solution set.

Equation A: 4/5(3f−7)=2/5(4f−8)


Equation B: 5/9g−2=7/9


Equation C: −2(4−2h)=−3(1−h)


(1 point)
Responses

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1 answer

To determine how many equations have the same solution set, we need to solve each equation and see if they yield the same solution.

Equation A: \[ \frac{4}{5}(3f - 7) = \frac{2}{5}(4f - 8) \]

  1. Multiply both sides by 5 to eliminate the fractions: \[ 4(3f - 7) = 2(4f - 8) \]
  2. Distribute: \[ 12f - 28 = 8f - 16 \]
  3. Bring all terms involving \(f\) to one side and constants to the other side: \[ 12f - 8f = -16 + 28 \] \[ 4f = 12 \]
  4. Solve for \(f\): \[ f = 3 \]

Equation B: \[ \frac{5}{9}g - 2 = \frac{7}{9} \]

  1. Add 2 to both sides: \[ \frac{5}{9}g = \frac{7}{9} + 2 \]
  2. Convert \(2\) to a fraction with a denominator of 9: \[ \frac{5}{9}g = \frac{7}{9} + \frac{18}{9} \] \[ \frac{5}{9}g = \frac{25}{9} \]
  3. Multiply by \(\frac{9}{5}\) to solve for \(g\): \[ g = 5 \]

Equation C: \[ -2(4 - 2h) = -3(1 - h) \]

  1. Distribute: \[ -8 + 4h = -3 + 3h \]
  2. Rearrange to find \(h\): \[ 4h - 3h = -3 + 8 \] \[ h = 5 \]

Now we will summarize the solutions:

  • Equation A has a solution \(f = 3\)
  • Equation B has a solution \(g = 5\)
  • Equation C has a solution \(h = 5\)

From our findings:

  • Equation A has a unique solution \(f=3\).
  • Equations B and C both have \(g=5\) and \(h=5\).

Conclusion: Based on our solutions, there is none that has the same solution set at the same solution values (since \(f\) and \(g\) or \(h\) do not match). There are 2 equations (B and C) that share a solution but \(A\) does not. Therefore, the answer is:

2