Solve frac{P_{1}V_{1}}{T_{1}}=frac{P_{2}V_{2}}{T_{2}}

$$\frac { P _ { 1 } V _ { 1 } } { T _ { 1 } } = \frac { P _ { 2 } V _ { 2 } } { T _ { 2 } }$$

Solve for P_1 (complex solution)

$$\left\{\begin{matrix}P_{1}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}\text{, }&T_{2}\neq 0\text{ and }V_{1}\neq 0\text{ and }T_{1}\neq 0\\P_{1}\in \mathrm{C}\text{, }&\left(V_{2}=0\text{ or }P_{2}=0\right)\text{ and }V_{1}=0\text{ and }T_{2}\neq 0\text{ and }T_{1}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by T_{1}T_{2}, the least common multiple of T_{1},T_{2}.

$$T_{2}P_{1}V_{1}=T_{1}P_{2}V_{2}$$

Reorder the terms.

$$P_{1}T_{2}V_{1}=P_{2}T_{1}V_{2}$$

The equation is in standard form.

$$T_{2}V_{1}P_{1}=P_{2}T_{1}V_{2}$$

Divide both sides by T_{2}V_{1}.

$$\frac{T_{2}V_{1}P_{1}}{T_{2}V_{1}}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}$$

Dividing by T_{2}V_{1} undoes the multiplication by T_{2}V_{1}.

$$P_{1}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}$$

Solve for P_2 (complex solution)

$$\left\{\begin{matrix}P_{2}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}\text{, }&T_{1}\neq 0\text{ and }V_{2}\neq 0\text{ and }T_{2}\neq 0\\P_{2}\in \mathrm{C}\text{, }&\left(V_{1}=0\text{ or }P_{1}=0\right)\text{ and }V_{2}=0\text{ and }T_{1}\neq 0\text{ and }T_{2}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by T_{1}T_{2}, the least common multiple of T_{1},T_{2}.

$$T_{2}P_{1}V_{1}=T_{1}P_{2}V_{2}$$

Swap sides so that all variable terms are on the left hand side.

$$T_{1}P_{2}V_{2}=T_{2}P_{1}V_{1}$$

The equation is in standard form.

$$T_{1}V_{2}P_{2}=P_{1}T_{2}V_{1}$$

Divide both sides by T_{1}V_{2}.

$$\frac{T_{1}V_{2}P_{2}}{T_{1}V_{2}}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}$$

Dividing by T_{1}V_{2} undoes the multiplication by T_{1}V_{2}.

$$P_{2}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}$$

Solve for P_1

$$\left\{\begin{matrix}P_{1}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}\text{, }&T_{2}\neq 0\text{ and }V_{1}\neq 0\text{ and }T_{1}\neq 0\\P_{1}\in \mathrm{R}\text{, }&\left(V_{2}=0\text{ or }P_{2}=0\right)\text{ and }V_{1}=0\text{ and }T_{2}\neq 0\text{ and }T_{1}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by T_{1}T_{2}, the least common multiple of T_{1},T_{2}.

$$T_{2}P_{1}V_{1}=T_{1}P_{2}V_{2}$$

Reorder the terms.

$$P_{1}T_{2}V_{1}=P_{2}T_{1}V_{2}$$

The equation is in standard form.

$$T_{2}V_{1}P_{1}=P_{2}T_{1}V_{2}$$

Divide both sides by T_{2}V_{1}.

$$\frac{T_{2}V_{1}P_{1}}{T_{2}V_{1}}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}$$

Dividing by T_{2}V_{1} undoes the multiplication by T_{2}V_{1}.

$$P_{1}=\frac{P_{2}T_{1}V_{2}}{T_{2}V_{1}}$$

Solve for P_2

$$\left\{\begin{matrix}P_{2}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}\text{, }&T_{1}\neq 0\text{ and }V_{2}\neq 0\text{ and }T_{2}\neq 0\\P_{2}\in \mathrm{R}\text{, }&\left(V_{1}=0\text{ or }P_{1}=0\right)\text{ and }V_{2}=0\text{ and }T_{1}\neq 0\text{ and }T_{2}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by T_{1}T_{2}, the least common multiple of T_{1},T_{2}.

$$T_{2}P_{1}V_{1}=T_{1}P_{2}V_{2}$$

Swap sides so that all variable terms are on the left hand side.

$$T_{1}P_{2}V_{2}=T_{2}P_{1}V_{1}$$

The equation is in standard form.

$$T_{1}V_{2}P_{2}=P_{1}T_{2}V_{1}$$

Divide both sides by T_{1}V_{2}.

$$\frac{T_{1}V_{2}P_{2}}{T_{1}V_{2}}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}$$

Dividing by T_{1}V_{2} undoes the multiplication by T_{1}V_{2}.

$$P_{2}=\frac{P_{1}T_{2}V_{1}}{T_{1}V_{2}}$$