(i) Start with the equation \(3^y = 4^{2-x}\).

Take the natural logarithm of both sides: \(\ln(3^y) = \ln(4^{2-x})\).

Using the logarithm power rule, this becomes \(y \ln 3 = (2-x) \ln 4\).

Rearrange to get \(y \ln 3 = 2 \ln 4 - x \ln 4\).

This can be written as \(y = -\frac{\ln 4}{\ln 3} x + \frac{2 \ln 4}{\ln 3}\), which is in the form \(y = mx + c\), showing it is a straight line.

The gradient \(m\) is \(-\frac{\ln 4}{\ln 3}\).

(ii) To find the intersection with \(y = 2x\), set \(-\frac{\ln 4}{\ln 3} x + \frac{2 \ln 4}{\ln 3} = 2x\).

Rearrange to get \(\frac{2 \ln 4}{\ln 3} = 2x + \frac{\ln 4}{\ln 3} x\).

Factor out \(x\): \(x \left(2 + \frac{\ln 4}{\ln 3}\right) = \frac{2 \ln 4}{\ln 3}\).

Solve for \(x\): \(x = \frac{\ln 4}{\ln 6}\).