Solve {l}{x+y+z=1}{2x+y-z=3}{x-y-z=1}

$$\left. \begin{array} { l } { x + y + z = 1 } \\ { 2 x + y - z = 3 } \\ { x - y - z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=\frac{1}{2}=0.5$$

$$z=-\frac{1}{2}=-0.5$$

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.

$$2\left(-y-z+1\right)+y-z=3 -y-z+1-y-z=1$$

Solve these equations for y and z respectively.

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

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

$$z=-\left(-3z-1\right)$$

Solve z=-\left(-3z-1\right) for z.

$$z=-\frac{1}{2}$$

Substitute -\frac{1}{2} for z in the equation y=-3z-1.

$$y=-3\left(-\frac{1}{2}\right)-1$$

Calculate y from y=-3\left(-\frac{1}{2}\right)-1.

$$y=\frac{1}{2}$$

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

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

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

$$x=1$$

The system is now solved.

$$x=1 y=\frac{1}{2} z=-\frac{1}{2}$$