Solve {l}{x+y+z=a}{x+y-z=b}{x-y+z=c}

$$\left\{ \begin{array} { l } { x + y + z = a } \\ { x + y - z = b } \\ { x - y + z = c } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{b+c}{2}$$

$$y=\frac{a-c}{2}$$

$$z=\frac{a-b}{2}$$

Short Steps Using Substitution

Solve x+y+z=a for x.

$$x=-y-z+a$$

Substitute -y-z+a for x in the second and third equation.

$$-y-z+a+y-z=b -y-z+a-y+z=c$$

Solve these equations for z and y respectively.

$$z=\frac{1}{2}a-\frac{1}{2}b y=\frac{1}{2}a-\frac{1}{2}c$$

Substitute \frac{1}{2}a-\frac{1}{2}b for z and \frac{1}{2}a-\frac{1}{2}c for y in the equation x=-y-z+a.

$$x=-\left(\frac{1}{2}a-\frac{1}{2}c\right)-\left(\frac{1}{2}a-\frac{1}{2}b\right)+a$$

Calculate x from x=-\left(\frac{1}{2}a-\frac{1}{2}c\right)-\left(\frac{1}{2}a-\frac{1}{2}b\right)+a.

$$x=\frac{1}{2}b+\frac{1}{2}c$$

The system is now solved.

$$x=\frac{1}{2}b+\frac{1}{2}c y=\frac{1}{2}a-\frac{1}{2}c z=\frac{1}{2}a-\frac{1}{2}b$$