Which of the following pairs of equations has exactly the same solution?
a. 3/8x=1 and 1/3x=1/8
b. −3.2x=0.64 and x4=−0.05
c. x3.2=1.8 and 1.8x=3.2
d. −3/4x=5/2 and 5/2x=−3/4
To find which pairs of equations have exactly the same solution, we need to solve each pair of equations and see if they lead to the same value of x.
a. Solve 3/8x=1:
Multiply both sides by 8/3:
x = 8/3
Solve 1/3x=1/8:
Multiply both sides by 8/1:
x = 8/1
x = 8
The solutions for x in the two equations are not the same, so this pair of equations does not have exactly the same solution.
b. Solve -3.2x=0.64:
Divide both sides by -3.2:
x = -0.20
Solve x^4 = -0.05:
We cannot solve this equation for x as it is a quartic equation and does not have a simple solution.
Since we cannot solve the second equation, we cannot compare the solutions. Therefore, we cannot determine if this pair of equations has exactly the same solution.
c. Solve x^3.2 = 1.8:
We cannot solve this equation directly as it involves a fractional exponent.
Solve 1.8x = 3.2:
Divide both sides by 1.8:
x = 3.2/1.8
x = 1.7778
Since we only have the solution for one equation in this pair, we cannot determine if they have exactly the same solution.
d. Solve -3/4x = 5/2:
Multiply both sides by -4/3:
x = (5/2)(-4/3)
x = -10/3
Solve 5/2x = -3/4:
Multiply both sides by 4/5:
x = (-3/4)(4/5)
x = -3/5
The solutions for x in both equations are the same, x = -10/3 and x = -3/5. Therefore, this pair of equations has exactly the same solution.
Therefore, the correct answer is option d. (-3/4x=5/2 and 5/2x=-3/4)