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

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

Solve for x, y, z

$$x = \frac{53}{19} = 2\frac{15}{19} \approx 2.789473684$$

$$y = -\frac{58}{19} = -3\frac{1}{19} \approx -3.052631579$$

$$z=\frac{7}{19}\approx 0.368421053$$

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.

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

Solve these equations for x and z respectively.

$$x=\frac{8}{3}+\frac{1}{3}z z=-\frac{4}{5}x+\frac{13}{5}$$

Substitute \frac{8}{3}+\frac{1}{3}z for x in the equation z=-\frac{4}{5}x+\frac{13}{5}.

$$z=-\frac{4}{5}\left(\frac{8}{3}+\frac{1}{3}z\right)+\frac{13}{5}$$

Solve z=-\frac{4}{5}\left(\frac{8}{3}+\frac{1}{3}z\right)+\frac{13}{5} for z.

$$z=\frac{7}{19}$$

Substitute \frac{7}{19} for z in the equation x=\frac{8}{3}+\frac{1}{3}z.

$$x=\frac{8}{3}+\frac{1}{3}\times \frac{7}{19}$$

Calculate x from x=\frac{8}{3}+\frac{1}{3}\times \frac{7}{19}.

$$x=\frac{53}{19}$$

Substitute \frac{53}{19} for x and \frac{7}{19} for z in the equation y=2x+z-9.

$$y=2\times \frac{53}{19}+\frac{7}{19}-9$$

Calculate y from y=2\times \frac{53}{19}+\frac{7}{19}-9.

$$y=-\frac{58}{19}$$

The system is now solved.

$$x=\frac{53}{19} y=-\frac{58}{19} z=\frac{7}{19}$$