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

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

Solve for x, y, z

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

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

$$z=-1$$

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.

$$-y-z+1-y+z=2 -y-z+1+y-z=3$$

Solve these equations for y and z respectively.

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

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

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

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

$$x=\frac{5}{2}$$

The system is now solved.

$$x=\frac{5}{2} y=-\frac{1}{2} z=-1$$