Solve {l}{2x+y+z=0}{3x-y+2z=11}{x-y+z=4}

$$\left. \begin{array} { l } { 2 x + y + z = 0 } \\ { 3 x - y + 2 z = 11 } \\ { x - y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=10$$

$$y=-7$$

$$z=-13$$

Short Steps Using Substitution

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

$$y=-2x-z$$

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

$$3x-\left(-2x-z\right)+2z=11 x-\left(-2x-z\right)+z=4$$

Solve these equations for x and z respectively.

$$x=-\frac{3}{5}z+\frac{11}{5} z=-\frac{3}{2}x+2$$

Substitute -\frac{3}{5}z+\frac{11}{5} for x in the equation z=-\frac{3}{2}x+2.

$$z=-\frac{3}{2}\left(-\frac{3}{5}z+\frac{11}{5}\right)+2$$

Solve z=-\frac{3}{2}\left(-\frac{3}{5}z+\frac{11}{5}\right)+2 for z.

$$z=-13$$

Substitute -13 for z in the equation x=-\frac{3}{5}z+\frac{11}{5}.

$$x=-\frac{3}{5}\left(-13\right)+\frac{11}{5}$$

Calculate x from x=-\frac{3}{5}\left(-13\right)+\frac{11}{5}.

$$x=10$$

Substitute 10 for x and -13 for z in the equation y=-2x-z.

$$y=-2\times 10-\left(-13\right)$$

Calculate y from y=-2\times 10-\left(-13\right).

$$y=-7$$

The system is now solved.

$$x=10 y=-7 z=-13$$