Solve {l}{2x+y+z=-3}{-2x+y+2z=4}{-x+2y+z=5}

$$\left. \begin{array}{l}{ 2 x + y + z = - 3 }\\{ - 2 x + y + 2 z = 4 }\\{ - x + 2 y + z = 5 }\end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=2$$

$$z=-1$$

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.

$$-2x-2x-z-3+2z=4 -x+2\left(-2x-z-3\right)+z=5$$

Solve these equations for x and z respectively.

$$x=-\frac{7}{4}+\frac{1}{4}z z=-5x-11$$

Substitute -\frac{7}{4}+\frac{1}{4}z for x in the equation z=-5x-11.

$$z=-5\left(-\frac{7}{4}+\frac{1}{4}z\right)-11$$

Solve z=-5\left(-\frac{7}{4}+\frac{1}{4}z\right)-11 for z.

$$z=-1$$

Substitute -1 for z in the equation x=-\frac{7}{4}+\frac{1}{4}z.

$$x=-\frac{7}{4}+\frac{1}{4}\left(-1\right)$$

Calculate x from x=-\frac{7}{4}+\frac{1}{4}\left(-1\right).

$$x=-2$$

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

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

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

$$y=2$$

The system is now solved.

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