Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary.

Instead: \begin{align}\frac VT=f(P)\\ PV=g(T),\tag1\end{align} where the functions $f$ and $g$ are not identically zero. (When two proportional variables are varying against each other, every other relevant variable in the system is fixed at a constant value.)

Thus, $$P\times f(P)=\frac{g(T)}T.$$ Since $P$ and $T$ are independent variables, $\big(P\times f(P)\big)$ and $\dfrac{g(T)}T$ must be a (nonzero) constant, say $k.$ Then, substituting $g(T)=kT$ into $(1)$ gives the combined gas law $$\frac{PV}{T}=k,$$ as required.

Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary.

Instead: \begin{align}\frac VT=f(P)\\ PV=g(T),\tag1\end{align} where the functions $f$ and $g$ are not identically zero. (When two proportional variables are varying against each other, every other relevant variable in the system is being fixed at a constant value.)

Thus, $$P\times f(P)=\frac{g(T)}T.$$ Since $P$ and $T$ are independent variables, $\big(P\times f(P)\big)$ and $\dfrac{g(T)}T$ must be a (nonzero) constant, say $k.$ Then, substituting $g(T)=kT$ into $(1)$ gives the combined gas law $$\frac{PV}{T}=k,$$ as required.

Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary.

Instead: \begin{align}\frac VT=f(P)\\ PV=g(T),\tag1\end{align} where the functions $f$ and $g$ are not identically zero.

Thus, $$P\times f(P)=\frac{g(T)}T.$$ Since $P$ and $T$ are independent variables, $\big(P\times f(P)\big)$ and $\dfrac{g(T)}T$ must be a (nonzero) constant, say $k.$ Then, substituting $g(T)=kT$ into $(1)$ gives the combined gas law $$\frac{PV}{T}=k,$$ as required.

Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary.

Instead: \begin{align}\frac VT=f(P)\\ PV=g(T),\tag1\end{align} where the functions $f$ and $g$ are not identically zero. (When two proportional variables are varying against each other, every other relevant variable in the system is fixed at a constant value.)

Thus, $$P\times f(P)=\frac{g(T)}T.$$ Since $P$ and $T$ are independent variables, $\big(P\times f(P)\big)$ and $\dfrac{g(T)}T$ must be a (nonzero) constant, say $k.$ Then, substituting $g(T)=kT$ into $(1)$ gives the combined gas law $$\frac{PV}{T}=k,$$ as required.