Solve {l}{2x-y+z=3}{-x+y+2z=5}{x-2y-z=0}

$$\left\{ \begin{array} { l } { 2 x - y + z = 3 } \\ { - x + y + 2 z = 5 } \\ { x - 2 y - z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-2$$

$$z=3$$

Short Steps Using Substitution

Solve 2x-y+z=3 for y.

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

Substitute 2x+z-3 for y in the second and third equation.

$$-x+2x+z-3+2z=5 x-2\left(2x+z-3\right)-z=0$$

Solve these equations for x and z respectively.

$$x=-3z+8 z=2-x$$

Substitute -3z+8 for x in the equation z=2-x.

$$z=2-\left(-3z+8\right)$$

Solve z=2-\left(-3z+8\right) for z.

$$z=3$$

Substitute 3 for z in the equation x=-3z+8.

$$x=-3\times 3+8$$

Calculate x from x=-3\times 3+8.

$$x=-1$$

Substitute -1 for x and 3 for z in the equation y=2x+z-3.

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

Calculate y from y=2\left(-1\right)+3-3.

$$y=-2$$

The system is now solved.

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