solve for all values of x and y that satisfy the following system of equations: 1=(x^1/2)(y^1/2) and ln(y)+1=2ln(x)

Question

anonymous · Answer

$$ 1 = \sqrt{x}\sqrt{y} = xy$$ (I take exp^2 in both sides)
Cause;  1^2 = (sqrtx sqrty)^2 = 1 = xy

Then  x = 1/y

$$\ln(y)+1=2\ln(1/y) $$ (cause x = 1/y)

$$1 = 2\ln(1/y) - \ln(y) = 2 \ln(1/y^2) $$

anonymous · Answer

Then take e^x at both sides

$$e^1 = e ^{2\ln(1/y^2)} = 2e ^{\ln(1/y^2)} = 2.1/y^2 = 2/y^2$$

anonymous · Answer

So  e = 2/y^2 Then solving for y;    y^2 = 2/e 

And  $$y = \sqrt{2/e}$$ then solve  for x