Which of the following pairs of equations has exactly the same solution?(1 point)

Question

Responses

−3.2x=0.64 and x/4=−0.05
negative 3.2x equals 0.64 and Start Fraction x over 4 End Fraction equals negative 0.05

−3/4x=5/2 and 5/2x=−3/4
negative Start Fraction 3 over 4 End Fraction x equals Start Fraction 5 over 2 End Fraction and Start Fraction 5 over 2 End Fraction x equals negative Start Fraction 3 over 4 End Fraction

x/3.2=1.8 and 1.8x=3.2
Start Fraction x over 3.2 End Fraction equals 1.8 and 1.8x equals 3.2

3/8x=1and 1/3x=18

Answer 1

To determine which pairs of equations have exactly the same solution, we can solve each pair or transform them into a simpler form and compare.

1. Equation Pair 1:

   • First Equation: \(-3.2x = 0.64\) \[ x = \frac{0.64}{-3.2} = -0.2 \]
   • Second Equation: \( \frac{x}{4} = -0.05 \) \[ x = -0.05 \times 4 = -0.2 \]

   Both equations yield \( x = -0.2 \).

2. Equation Pair 2:

   • First Equation: \(-\frac{3}{4}x = \frac{5}{2}\) \[ x = -\frac{5/2}{3/4} = -\frac{5 \times 4}{2 \times 3} = -\frac{20}{6} = -\frac{10}{3} \]
   • Second Equation: \(\frac{5}{2}x = -\frac{3}{4}\) \[ x = -\frac{3/4}{5/2} = -\frac{3 \times 2}{4 \times 5} = -\frac{6}{20} = -\frac{3}{10} \]

   These equations do not have the same solution.

3. Equation Pair 3:

   • First Equation: \(\frac{x}{3.2} = 1.8\) \[ x = 1.8 \times 3.2 = 5.76 \]
   • Second Equation: \(1.8x = 3.2\) \[ x = \frac{3.2}{1.8} = \frac{32}{18} = \frac{16}{9} \]

   These equations do not have the same solution.

4. Equation Pair 4:

   • First Equation: \(\frac{3}{8}x = 1\) \[ x = \frac{1}{(3/8)} = \frac{8}{3} \]
   • Second Equation: \(\frac{1}{3}x = 18\) \[ x = 18 \times 3 = 54 \]

   These equations do not have the same solution.

Thus, the only pair with exactly the same solution is Equation Pair 1:

\(-3.2x = 0.64\) and \(\frac{x}{4} = -0.05\).