Løs m_{1}*v_{1}+m_{2}*v_{2}=(m_{1}+m_{2})*v_{g}

$$m _ { 1 } \cdot v _ { 1 } + m _ { 2 } \cdot v _ { 2 } = ( m _ { 1 } + m _ { 2 } ) \cdot v _ { g }$$

Løs for m_1

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

Vejledning i at løse lineær ligning

Brug fordelingsegenskaben til at multiplicere m_{1}+m_{2} med v_{g}.

$$m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{g}+m_{2}v_{g}$$

Subtraher m_{1}v_{g} fra begge sider.

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

Subtraher m_{2}v_{2} fra begge sider.

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

Kombiner alle led med m_{1}.

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

Divider begge sider med v_{1}-v_{g}.

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

Division med v_{1}-v_{g} annullerer multiplikationen med v_{1}-v_{g}.

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

Løs for m_2

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

Vejledning i at løse lineær ligning

Brug fordelingsegenskaben til at multiplicere m_{1}+m_{2} med v_{g}.

$$m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{g}+m_{2}v_{g}$$

Subtraher m_{2}v_{g} fra begge sider.

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

Subtraher m_{1}v_{1} fra begge sider.

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

Kombiner alle led med m_{2}.

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

Divider begge sider med v_{2}-v_{g}.

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

Division med v_{2}-v_{g} annullerer multiplikationen med v_{2}-v_{g}.

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