Solve {l}{x-2y+z=0}{2x-y+5z=-3}{3x+y+2z=1}

$$\left\{ \begin{array} { l } { x - 2 y + z = 0 } \\ { 2 x - y + 5 z = - 3 } \\ { 3 x + y + 2 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=0$$

$$z=-1$$

Short Steps Using Substitution

Solve x-2y+z=0 for x.

$$x=2y-z$$

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

$$2\left(2y-z\right)-y+5z=-3 3\left(2y-z\right)+y+2z=1$$

Solve these equations for y and z respectively.

$$y=-1-z z=7y-1$$

Substitute -1-z for y in the equation z=7y-1.

$$z=7\left(-1-z\right)-1$$

Solve z=7\left(-1-z\right)-1 for z.

$$z=-1$$

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

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

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

$$y=0$$

Substitute 0 for y and -1 for z in the equation x=2y-z.

$$x=2\times 0-\left(-1\right)$$

Calculate x from x=2\times 0-\left(-1\right).

$$x=1$$

The system is now solved.

$$x=1 y=0 z=-1$$