Solve {l}{x+y+z=6}{x-y-z=4}{2x-y+z=1}

$$\left\{ \begin{array} { l } { x + y + z = 6 } \\ { x - y - z = 4 } \\ { 2 x - y + z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=5$$

$$z=-4$$

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.

$$-y-z+6-y-z=4 2\left(-y-z+6\right)-y+z=1$$

Solve these equations for y and z respectively.

$$y=1-z z=-3y+11$$

Substitute 1-z for y in the equation z=-3y+11.

$$z=-3\left(1-z\right)+11$$

Solve z=-3\left(1-z\right)+11 for z.

$$z=-4$$

Substitute -4 for z in the equation y=1-z.

$$y=1-\left(-4\right)$$

Calculate y from y=1-\left(-4\right).

$$y=5$$

Substitute 5 for y and -4 for z in the equation x=-y-z+6.

$$x=-5-\left(-4\right)+6$$

Calculate x from x=-5-\left(-4\right)+6.

$$x=5$$

The system is now solved.

$$x=5 y=5 z=-4$$