Solve {l}{x+y+z=1}{x-y-z=5}{z-y-x=1}

$$\left. \begin{array} { l } { x + y + z = 1 } \\ { x - y - z = 5 } \\ { z - y - x = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-3$$

$$z=1$$

Short Steps Using Substitution

Solve x+y+z=1 for x.

$$x=-y-z+1$$

Substitute -y-z+1 for x in the second and third equation.

$$-y-z+1-y-z=5 z-y-\left(-y-z+1\right)=1$$

Solve these equations for y and z respectively.

$$y=-2-z z=1$$

Substitute 1 for z in the equation y=-2-z.

$$y=-2-1$$

Calculate y from y=-2-1.

$$y=-3$$

Substitute -3 for y and 1 for z in the equation x=-y-z+1.

$$x=-\left(-3\right)-1+1$$

Calculate x from x=-\left(-3\right)-1+1.

$$x=3$$

The system is now solved.

$$x=3 y=-3 z=1$$