Solve {l}{2x+y-z=9}{x-2y+z=-1}{x+2y+2z=-5}

$$\left. \begin{array}{l}{ 2 x + y - z = 9 }\\{ x - 2 y + z = - 1 }\\{ x + 2 y + 2 z = - 5 }\end{array} \right.$$

Solve for x, y, z

$$x = \frac{45}{17} = 2\frac{11}{17} \approx 2.647058824$$

$$y=-\frac{1}{17}\approx -0.058823529$$

$$z = -\frac{64}{17} = -3\frac{13}{17} \approx -3.764705882$$

Short Steps Using Substitution

Solve 2x+y-z=9 for y.

$$y=-2x+z+9$$

Substitute -2x+z+9 for y in the second and third equation.

$$x-2\left(-2x+z+9\right)+z=-1 x+2\left(-2x+z+9\right)+2z=-5$$

Solve these equations for x and z respectively.

$$x=\frac{17}{5}+\frac{1}{5}z z=\frac{3}{4}x-\frac{23}{4}$$

Substitute \frac{17}{5}+\frac{1}{5}z for x in the equation z=\frac{3}{4}x-\frac{23}{4}.

$$z=\frac{3}{4}\left(\frac{17}{5}+\frac{1}{5}z\right)-\frac{23}{4}$$

Solve z=\frac{3}{4}\left(\frac{17}{5}+\frac{1}{5}z\right)-\frac{23}{4} for z.

$$z=-\frac{64}{17}$$

Substitute -\frac{64}{17} for z in the equation x=\frac{17}{5}+\frac{1}{5}z.

$$x=\frac{17}{5}+\frac{1}{5}\left(-\frac{64}{17}\right)$$

Calculate x from x=\frac{17}{5}+\frac{1}{5}\left(-\frac{64}{17}\right).

$$x=\frac{45}{17}$$

Substitute \frac{45}{17} for x and -\frac{64}{17} for z in the equation y=-2x+z+9.

$$y=-2\times \frac{45}{17}-\frac{64}{17}+9$$

Calculate y from y=-2\times \frac{45}{17}-\frac{64}{17}+9.

$$y=-\frac{1}{17}$$

The system is now solved.

$$x=\frac{45}{17} y=-\frac{1}{17} z=-\frac{64}{17}$$