Solve {l}{x+2y-z=8}{2x+2y-z=10}{2x+3y+z=5}

$$\left\{ \begin{array} { l } { x + 2 y - z = 8 } \\ { 2 x + 2 y - z = 10 } \\ { 2 x + 3 y + z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

$$z = -\frac{16}{5} = -3\frac{1}{5} = -3.2$$

Short Steps Using Substitution

Solve x+2y-z=8 for x.

$$x=-2y+z+8$$

Substitute -2y+z+8 for x in the second and third equation.

$$2\left(-2y+z+8\right)+2y-z=10 2\left(-2y+z+8\right)+3y+z=5$$

Solve these equations for y and z respectively.

$$y=\frac{1}{2}z+3 z=-\frac{11}{3}+\frac{1}{3}y$$

Substitute \frac{1}{2}z+3 for y in the equation z=-\frac{11}{3}+\frac{1}{3}y.

$$z=-\frac{11}{3}+\frac{1}{3}\left(\frac{1}{2}z+3\right)$$

Solve z=-\frac{11}{3}+\frac{1}{3}\left(\frac{1}{2}z+3\right) for z.

$$z=-\frac{16}{5}$$

Substitute -\frac{16}{5} for z in the equation y=\frac{1}{2}z+3.

$$y=\frac{1}{2}\left(-\frac{16}{5}\right)+3$$

Calculate y from y=\frac{1}{2}\left(-\frac{16}{5}\right)+3.

$$y=\frac{7}{5}$$

Substitute \frac{7}{5} for y and -\frac{16}{5} for z in the equation x=-2y+z+8.

$$x=-2\times \frac{7}{5}-\frac{16}{5}+8$$

Calculate x from x=-2\times \frac{7}{5}-\frac{16}{5}+8.

$$x=2$$

The system is now solved.

$$x=2 y=\frac{7}{5} z=-\frac{16}{5}$$