Solve alpha+beta+gamma=valphabetagamma

$$\alpha + \beta + \gamma = v \alpha \beta \gamma$$

Solve for v (complex solution)

$$\left\{\begin{matrix}v=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }\text{, }&\gamma \neq 0\text{ and }\beta \neq 0\text{ and }\alpha \neq 0\\v\in \mathrm{C}\text{, }&\left(\alpha =-\gamma \text{ and }\beta =0\right)\text{ or }\left(\alpha =-\beta \text{ and }\gamma =0\right)\text{ or }\left(\alpha =0\text{ and }\beta =-\gamma \text{ and }\gamma \neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v\alpha \beta \gamma =\alpha +\beta +\gamma$$

The equation is in standard form.

$$\alpha \beta \gamma v=\alpha +\beta +\gamma$$

Divide both sides by \alpha \beta \gamma .

$$\frac{\alpha \beta \gamma v}{\alpha \beta \gamma }=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }$$

Dividing by \alpha \beta \gamma undoes the multiplication by \alpha \beta \gamma .

$$v=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }$$

Solve for α (complex solution)

$$\left\{\begin{matrix}\alpha =-\frac{\beta +\gamma }{1-v\beta \gamma }\text{, }&\gamma =0\text{ or }v=0\text{ or }\beta \neq \frac{1}{v\gamma }\\\alpha \in \mathrm{C}\text{, }&\left(\gamma =-iv^{-\frac{1}{2}}\text{ and }\beta =iv^{-\frac{1}{2}}\text{ and }v\neq 0\right)\text{ or }\left(\gamma =iv^{-\frac{1}{2}}\text{ and }\beta =-iv^{-\frac{1}{2}}\text{ and }v\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract v\alpha \beta \gamma from both sides.

$$\alpha +\beta +\gamma -v\alpha \beta \gamma =0$$

Subtract \beta from both sides. Anything subtracted from zero gives its negation.

$$\alpha +\gamma -v\alpha \beta \gamma =-\beta$$

Subtract \gamma from both sides.

$$\alpha -v\alpha \beta \gamma =-\beta -\gamma$$

Combine all terms containing \alpha .

$$\left(1-v\beta \gamma \right)\alpha =-\beta -\gamma$$

Divide both sides by 1-v\beta \gamma .

$$\frac{\left(1-v\beta \gamma \right)\alpha }{1-v\beta \gamma }=\frac{-\beta -\gamma }{1-v\beta \gamma }$$

Dividing by 1-v\beta \gamma undoes the multiplication by 1-v\beta \gamma .

$$\alpha =\frac{-\beta -\gamma }{1-v\beta \gamma }$$

Divide -\beta -\gamma by 1-v\beta \gamma .

$$\alpha =-\frac{\beta +\gamma }{1-v\beta \gamma }$$

Solve for v

$$\left\{\begin{matrix}v=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }\text{, }&\gamma \neq 0\text{ and }\beta \neq 0\text{ and }\alpha \neq 0\\v\in \mathrm{R}\text{, }&\left(\alpha =-\gamma \text{ and }\beta =0\right)\text{ or }\left(\alpha =-\beta \text{ and }\gamma =0\right)\text{ or }\left(\alpha =0\text{ and }\beta =-\gamma \text{ and }\gamma \neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v\alpha \beta \gamma =\alpha +\beta +\gamma$$

The equation is in standard form.

$$\alpha \beta \gamma v=\alpha +\beta +\gamma$$

Divide both sides by \alpha \beta \gamma .

$$\frac{\alpha \beta \gamma v}{\alpha \beta \gamma }=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }$$

Dividing by \alpha \beta \gamma undoes the multiplication by \alpha \beta \gamma .

$$v=\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }$$

Solve for α

$$\left\{\begin{matrix}\alpha =-\frac{\beta +\gamma }{1-v\beta \gamma }\text{, }&\gamma =0\text{ or }v=0\text{ or }\beta \neq \frac{1}{v\gamma }\\\alpha \in \mathrm{R}\text{, }&\beta =-\gamma \text{ and }v=-\frac{1}{\gamma ^{2}}\text{ and }\gamma \neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract v\alpha \beta \gamma from both sides.

$$\alpha +\beta +\gamma -v\alpha \beta \gamma =0$$

Subtract \beta from both sides. Anything subtracted from zero gives its negation.

$$\alpha +\gamma -v\alpha \beta \gamma =-\beta$$

Subtract \gamma from both sides.

$$\alpha -v\alpha \beta \gamma =-\beta -\gamma$$

Combine all terms containing \alpha .

$$\left(1-v\beta \gamma \right)\alpha =-\beta -\gamma$$

Divide both sides by 1-v\beta \gamma .

$$\frac{\left(1-v\beta \gamma \right)\alpha }{1-v\beta \gamma }=\frac{-\beta -\gamma }{1-v\beta \gamma }$$

Dividing by 1-v\beta \gamma undoes the multiplication by 1-v\beta \gamma .

$$\alpha =\frac{-\beta -\gamma }{1-v\beta \gamma }$$

Divide -\beta -\gamma by 1-v\beta \gamma .

$$\alpha =-\frac{\beta +\gamma }{1-v\beta \gamma }$$