Solve m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}

$$m _ { 1 } u _ { 1 } + m _ { 2 } u _ { 2 } = m _ { 1 } v _ { 1 } + m _ { 2 } v _ { 2 }$$

Solve for m_1 (complex solution)

$$\left\{\begin{matrix}m_{1}=-\frac{m_{2}\left(u_{2}-v_{2}\right)}{u_{1}-v_{1}}\text{, }&u_{1}\neq v_{1}\\m_{1}\in \mathrm{C}\text{, }&\left(u_{2}=v_{2}\text{ or }m_{2}=0\right)\text{ and }u_{1}=v_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m_{1}v_{1} from both sides.

$$m_{1}u_{1}+m_{2}u_{2}-m_{1}v_{1}=m_{2}v_{2}$$

Subtract m_{2}u_{2} from both sides.

$$m_{1}u_{1}-m_{1}v_{1}=m_{2}v_{2}-m_{2}u_{2}$$

Combine all terms containing m_{1}.

$$\left(u_{1}-v_{1}\right)m_{1}=m_{2}v_{2}-m_{2}u_{2}$$

Divide both sides by u_{1}-v_{1}.

$$\frac{\left(u_{1}-v_{1}\right)m_{1}}{u_{1}-v_{1}}=\frac{m_{2}\left(v_{2}-u_{2}\right)}{u_{1}-v_{1}}$$

Dividing by u_{1}-v_{1} undoes the multiplication by u_{1}-v_{1}.

$$m_{1}=\frac{m_{2}\left(v_{2}-u_{2}\right)}{u_{1}-v_{1}}$$

Solve for m_2 (complex solution)

$$\left\{\begin{matrix}m_{2}=-\frac{m_{1}\left(u_{1}-v_{1}\right)}{u_{2}-v_{2}}\text{, }&u_{2}\neq v_{2}\\m_{2}\in \mathrm{C}\text{, }&\left(u_{1}=v_{1}\text{ or }m_{1}=0\right)\text{ and }u_{2}=v_{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m_{2}v_{2} from both sides.

$$m_{1}u_{1}+m_{2}u_{2}-m_{2}v_{2}=m_{1}v_{1}$$

Subtract m_{1}u_{1} from both sides.

$$m_{2}u_{2}-m_{2}v_{2}=m_{1}v_{1}-m_{1}u_{1}$$

Combine all terms containing m_{2}.

$$\left(u_{2}-v_{2}\right)m_{2}=m_{1}v_{1}-m_{1}u_{1}$$

Divide both sides by u_{2}-v_{2}.

$$\frac{\left(u_{2}-v_{2}\right)m_{2}}{u_{2}-v_{2}}=\frac{m_{1}\left(v_{1}-u_{1}\right)}{u_{2}-v_{2}}$$

Dividing by u_{2}-v_{2} undoes the multiplication by u_{2}-v_{2}.

$$m_{2}=\frac{m_{1}\left(v_{1}-u_{1}\right)}{u_{2}-v_{2}}$$

Solve for m_1

$$\left\{\begin{matrix}m_{1}=-\frac{m_{2}\left(u_{2}-v_{2}\right)}{u_{1}-v_{1}}\text{, }&u_{1}\neq v_{1}\\m_{1}\in \mathrm{R}\text{, }&\left(u_{2}=v_{2}\text{ or }m_{2}=0\right)\text{ and }u_{1}=v_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m_{1}v_{1} from both sides.

$$m_{1}u_{1}+m_{2}u_{2}-m_{1}v_{1}=m_{2}v_{2}$$

Subtract m_{2}u_{2} from both sides.

$$m_{1}u_{1}-m_{1}v_{1}=m_{2}v_{2}-m_{2}u_{2}$$

Combine all terms containing m_{1}.

$$\left(u_{1}-v_{1}\right)m_{1}=m_{2}v_{2}-m_{2}u_{2}$$

Divide both sides by u_{1}-v_{1}.

$$\frac{\left(u_{1}-v_{1}\right)m_{1}}{u_{1}-v_{1}}=\frac{m_{2}\left(v_{2}-u_{2}\right)}{u_{1}-v_{1}}$$

Dividing by u_{1}-v_{1} undoes the multiplication by u_{1}-v_{1}.

$$m_{1}=\frac{m_{2}\left(v_{2}-u_{2}\right)}{u_{1}-v_{1}}$$

Solve for m_2

$$\left\{\begin{matrix}m_{2}=-\frac{m_{1}\left(u_{1}-v_{1}\right)}{u_{2}-v_{2}}\text{, }&u_{2}\neq v_{2}\\m_{2}\in \mathrm{R}\text{, }&\left(u_{1}=v_{1}\text{ or }m_{1}=0\right)\text{ and }u_{2}=v_{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m_{2}v_{2} from both sides.

$$m_{1}u_{1}+m_{2}u_{2}-m_{2}v_{2}=m_{1}v_{1}$$

Subtract m_{1}u_{1} from both sides.

$$m_{2}u_{2}-m_{2}v_{2}=m_{1}v_{1}-m_{1}u_{1}$$

Combine all terms containing m_{2}.

$$\left(u_{2}-v_{2}\right)m_{2}=m_{1}v_{1}-m_{1}u_{1}$$

Divide both sides by u_{2}-v_{2}.

$$\frac{\left(u_{2}-v_{2}\right)m_{2}}{u_{2}-v_{2}}=\frac{m_{1}\left(v_{1}-u_{1}\right)}{u_{2}-v_{2}}$$

Dividing by u_{2}-v_{2} undoes the multiplication by u_{2}-v_{2}.

$$m_{2}=\frac{m_{1}\left(v_{1}-u_{1}\right)}{u_{2}-v_{2}}$$