Answer: \(x=\frac58\), \(y=\mathrm e^{3/8}\).

We start with the main method. Use logarithm laws and index laws first so that the equation is reduced to an algebraic equation.

The equations are

\(5x-3\ln y=2\)

and

\(x+\ln y=1.\)

Multiply the second equation by \(3\):

\(3x+3\ln y=3.\)

Now add this to the first equation:

\((5x-3\ln y)+(3x+3\ln y)=2+3.\)

So

\(8x=5,\)

and hence

\(x=\frac58.\)

Substitute this into \(x+\ln y=1\):

\(\frac58+\ln y=1.\)

Therefore

\(\ln y=\frac38.\)

So

\(y=\mathrm e^{3/8}.\)

This completes the solution and gives the required result.