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Having a contradiction while deriving Gay Lussac's-Lussac's law using Boyle's law and Charles' law

From Boyle's Lawlaw: $$\begin{aligned}P_1V_1&=P_2V_2\\ V_2&=\dfrac{P_1V_1}{P_2}\ \ \ \ \ \ \ \ \ (1)\end{aligned}$$

$$P_1V_1 = P_2V_2 \quad\implies\quad V_2 = \frac{P_1V_1}{P_2} \label{eqn:boyle}\tag{1}$$

From Charles' Lawlaw: $$\begin{aligned}\dfrac{V_1}{T_1}&=\dfrac{V_2}{T_2} \\ V_2&=\dfrac{V_1T_2}{T_1}\ \ \ \ \ \ \ \ \ (2)\end{aligned}$$$$\frac{V_1}{T_1} = \frac{V_2}{T_2} \quad\implies\quad V_2 = \frac{V_1T_2}{T_1} \label{eqn:charles}\tag{2}$$

Equating 1\eqref{eqn:boyle} and 2\eqref{eqn:charles} gives: $$\dfrac{P_1V_1}{P_2} = \dfrac{V_1T_2}{T_1}$$

$$\frac{P_1V_1}{P_2} = \frac{V_1T_2}{T_1},$$

which upon solving will give: $$P_1T_1=P_2T_2$$

$$P_1T_1 = P_2T_2,$$

which is exact opposite of Gay Lussac's Law-Lussac's law. I can't find the flaw in my logic. Help!

Having a contradiction while deriving Gay Lussac's law using Boyle's law and Charles' law

From Boyle's Law: $$\begin{aligned}P_1V_1&=P_2V_2\\ V_2&=\dfrac{P_1V_1}{P_2}\ \ \ \ \ \ \ \ \ (1)\end{aligned}$$

From Charles' Law: $$\begin{aligned}\dfrac{V_1}{T_1}&=\dfrac{V_2}{T_2} \\ V_2&=\dfrac{V_1T_2}{T_1}\ \ \ \ \ \ \ \ \ (2)\end{aligned}$$

Equating 1 and 2 gives: $$\dfrac{P_1V_1}{P_2} = \dfrac{V_1T_2}{T_1}$$

which upon solving will give: $$P_1T_1=P_2T_2$$

which is exact opposite of Gay Lussac's Law. I can't find the flaw in my logic. Help!

Having a contradiction while deriving Gay-Lussac's law using Boyle's law and Charles' law

From Boyle's law:

$$P_1V_1 = P_2V_2 \quad\implies\quad V_2 = \frac{P_1V_1}{P_2} \label{eqn:boyle}\tag{1}$$

From Charles' law: $$\frac{V_1}{T_1} = \frac{V_2}{T_2} \quad\implies\quad V_2 = \frac{V_1T_2}{T_1} \label{eqn:charles}\tag{2}$$

Equating \eqref{eqn:boyle} and \eqref{eqn:charles} gives:

$$\frac{P_1V_1}{P_2} = \frac{V_1T_2}{T_1},$$

which upon solving will give:

$$P_1T_1 = P_2T_2,$$

which is exact opposite of Gay-Lussac's law. I can't find the flaw in my logic. Help!

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P1V1=P2V2(Boyle's law) V2=P1V1/P2.....(1) V1/T1=V2/T2(Charles' law)From Boyle's Law: V2=V1T2/T1.....(2)$$\begin{aligned}P_1V_1&=P_2V_2\\ V_2&=\dfrac{P_1V_1}{P_2}\ \ \ \ \ \ \ \ \ (1)\end{aligned}$$

From Charles' Law: Equating$$\begin{aligned}\dfrac{V_1}{T_1}&=\dfrac{V_2}{T_2} \\ V_2&=\dfrac{V_1T_2}{T_1}\ \ \ \ \ \ \ \ \ (2)\end{aligned}$$

Equating 1 and 2 gives  :- P1V1/P2=V1T2/T1 which$$\dfrac{P_1V_1}{P_2} = \dfrac{V_1T_2}{T_1}$$

which upon solving will give P1T1=P2T2: $$P_1T_1=P_2T_2$$

which is exact opposite of gay lussac's lawGay Lussac's Law. I can't find the flaw in my logic. Help!

P1V1=P2V2(Boyle's law) V2=P1V1/P2.....(1) V1/T1=V2/T2(Charles' law) V2=V1T2/T1.....(2) Equating 1 and 2 gives  :- P1V1/P2=V1T2/T1 which upon solving will give P1T1=P2T2 which is exact opposite of gay lussac's law. I can't find the flaw in my logic. Help!

From Boyle's Law: $$\begin{aligned}P_1V_1&=P_2V_2\\ V_2&=\dfrac{P_1V_1}{P_2}\ \ \ \ \ \ \ \ \ (1)\end{aligned}$$

From Charles' Law: $$\begin{aligned}\dfrac{V_1}{T_1}&=\dfrac{V_2}{T_2} \\ V_2&=\dfrac{V_1T_2}{T_1}\ \ \ \ \ \ \ \ \ (2)\end{aligned}$$

Equating 1 and 2 gives: $$\dfrac{P_1V_1}{P_2} = \dfrac{V_1T_2}{T_1}$$

which upon solving will give: $$P_1T_1=P_2T_2$$

which is exact opposite of Gay Lussac's Law. I can't find the flaw in my logic. Help!

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Having a contradiction while deriving Gay Lussac's law using Boyle's law and Charles' law

P1V1=P2V2(Boyle's law) V2=P1V1/P2.....(1) V1/T1=V2/T2(Charles' law) V2=V1T2/T1.....(2) Equating 1 and 2 gives :- P1V1/P2=V1T2/T1 which upon solving will give P1T1=P2T2 which is exact opposite of gay lussac's law. I can't find the flaw in my logic. Help!