Making y = 3x+5y=3x+5 we have
3^((y-5)/3) = y3y−53=y or
3^(y/3)3^(-5/3) = y3y33−53=y
now making 3^(1/3) = e^lambda313=eλ we have
e^(lambda y) 3^(-5/3) = yeλy3−53=y or
3^(-5/3)=y e^(-lambda y)3−53=ye−λy or
(-lambda)3^(-5/3) = (-lambda y) e^(-lambda y)(−λ)3−53=(−λy)e−λy
At this point we use the Lambert function
https://en.wikipedia.org/wiki/Lambert_W_function
Y = X e^X hArr X = W(Y)Y=XeX⇔X=W(Y) and then
-lambda y = W((-lambda)3^(-5/3))−λy=W((−λ)3−53) or
y = -1/lambda W((-lambda)3^(-5/3)) = 3x-5y=−1λW((−λ)3−53)=3x−5 and finally
x = (-5 Log 3 - 3 W(-(Log 3/(9 xx 3^(2/3)))))/(3 Log 3)x=−5log3−3W(−(log39×323))3log3
We have then two solutions
x =-1.60981x=−1.60981 and x = 2.24048x=2.24048
Attached a plot showing the intersections of y = 3^xy=3x and y = 3x+5y=3x+5