Solve m_{2}v_{0}=m_{1}w_{1}+m_{2}w_{2}

$$m _ { 2 } v _ { 0 } = m _ { 1 } w _ { 1 } + m _ { 2 } w _ { 2 }$$

Solve for m_1

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

Steps for Solving Linear Equation

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

$$m_{1}w_{1}+m_{2}w_{2}=m_{2}v_{0}$$

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

$$m_{1}w_{1}=m_{2}v_{0}-m_{2}w_{2}$$

The equation is in standard form.

$$w_{1}m_{1}=m_{2}v_{0}-m_{2}w_{2}$$

Divide both sides by w_{1}.

$$\frac{w_{1}m_{1}}{w_{1}}=\frac{m_{2}\left(v_{0}-w_{2}\right)}{w_{1}}$$

Dividing by w_{1} undoes the multiplication by w_{1}.

$$m_{1}=\frac{m_{2}\left(v_{0}-w_{2}\right)}{w_{1}}$$

Solve for m_2

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

Steps for Solving Linear Equation

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

$$m_{2}v_{0}-m_{2}w_{2}=m_{1}w_{1}$$

Combine all terms containing m_{2}.

$$\left(v_{0}-w_{2}\right)m_{2}=m_{1}w_{1}$$

Divide both sides by v_{0}-w_{2}.

$$\frac{\left(v_{0}-w_{2}\right)m_{2}}{v_{0}-w_{2}}=\frac{m_{1}w_{1}}{v_{0}-w_{2}}$$

Dividing by v_{0}-w_{2} undoes the multiplication by v_{0}-w_{2}.

$$m_{2}=\frac{m_{1}w_{1}}{v_{0}-w_{2}}$$