Question

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)

Answer 1

\[ 1 = \sqrt{x}\sqrt{y} = xy\] (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)

\[1 = 2\ln(1/y) - \ln(y) = 2 \ln(1/y^2) \]

Answer 2

Then take e^x at both sides

\[e^1 = e ^{2\ln(1/y^2)} = 2e ^{\ln(1/y^2)} = 2.1/y^2 = 2/y^2\]

Answer 3

So e = 2/y^2 Then solving for y; y^2 = 2/e

And \[y = \sqrt{2/e}\] then solve for x