Solve {l}{x+y+z=6}{x-y+z=2}{2x+y-z=1}

$$\left\{ \begin{array} { l } { x + y + z = 6 } \\ { x - y + z = 2 } \\ { 2 x + y - z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

$$x=-y-z+6$$

Substitute -y-z+6 for x in the second and third equation.

$$-y-z+6-y+z=2 2\left(-y-z+6\right)+y-z=1$$

Solve these equations for y and z respectively.

$$y=2 z=\frac{11}{3}-\frac{1}{3}y$$

Substitute 2 for y in the equation z=\frac{11}{3}-\frac{1}{3}y.

$$z=\frac{11}{3}-\frac{1}{3}\times 2$$

Calculate z from z=\frac{11}{3}-\frac{1}{3}\times 2.

$$z=3$$

Substitute 2 for y and 3 for z in the equation x=-y-z+6.

$$x=-2-3+6$$

Calculate x from x=-2-3+6.

$$x=1$$

The system is now solved.

$$x=1 y=2 z=3$$