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}.

$$p_{2}=\frac{p_{1}t_{2}v_{1}}{t_{1}v_{2}}$$