Solve {l}{x+y=alpha}{x+z=beta}{y+z=gamma}

$$\left\{ \begin{array} { l } { x + y = \alpha } \\ { x + z = \beta } \\ { y + z = \gamma } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{\alpha +\beta -\gamma }{2}$$

$$y=\frac{\alpha -\beta +\gamma }{2}$$

$$z=\frac{\gamma +\beta -\alpha }{2}$$

Short Steps Using Substitution

Solve x+y=\alpha for x.

$$x=-y+\alpha$$

Substitute -y+\alpha for x in the equation x+z=\beta .

$$-y+\alpha +z=\beta$$

Solve the second equation for y and the third equation for z.

$$y=\alpha +z-\beta z=-y+\gamma$$

Substitute \alpha +z-\beta for y in the equation z=-y+\gamma .

$$z=-\left(\alpha +z-\beta \right)+\gamma$$

Solve z=-\left(\alpha +z-\beta \right)+\gamma for z.

$$z=-\frac{1}{2}\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma$$

Substitute -\frac{1}{2}\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma for z in the equation y=\alpha +z-\beta .

$$y=\alpha -\frac{1}{2}\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma -\beta$$

Calculate y from y=\alpha -\frac{1}{2}\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma -\beta .

$$y=\frac{1}{2}\alpha -\frac{1}{2}\beta +\frac{1}{2}\gamma$$

Substitute \frac{1}{2}\alpha -\frac{1}{2}\beta +\frac{1}{2}\gamma for y in the equation x=-y+\alpha .

$$x=-\left(\frac{1}{2}\alpha -\frac{1}{2}\beta +\frac{1}{2}\gamma \right)+\alpha$$

Calculate x from x=-\left(\frac{1}{2}\alpha -\frac{1}{2}\beta +\frac{1}{2}\gamma \right)+\alpha .

$$x=\frac{1}{2}\alpha +\frac{1}{2}\beta -\frac{1}{2}\gamma$$

The system is now solved.

$$x=\frac{1}{2}\alpha +\frac{1}{2}\beta -\frac{1}{2}\gamma y=\frac{1}{2}\alpha -\frac{1}{2}\beta +\frac{1}{2}\gamma z=-\frac{1}{2}\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma$$