Solve {l}{x-2y+z=4}{x-y-4z=1}{x-4y+2z=8}

$$\left. \begin{array} { l } { x - 2 y + z = 4 } \\ { x - y - 4 z = 1 } \\ { x - 4 y + 2 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y = -\frac{17}{9} = -1\frac{8}{9} \approx -1.888888889$$

$$z=\frac{2}{9}\approx 0.222222222$$

Short Steps Using Substitution

Solve x-2y+z=4 for x.

$$x=2y-z+4$$

Substitute 2y-z+4 for x in the second and third equation.

$$2y-z+4-y-4z=1 2y-z+4-4y+2z=8$$

Solve these equations for y and z respectively.

$$y=5z-3 z=4+2y$$

Substitute 5z-3 for y in the equation z=4+2y.

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

Solve z=4+2\left(5z-3\right) for z.

$$z=\frac{2}{9}$$

Substitute \frac{2}{9} for z in the equation y=5z-3.

$$y=5\times \frac{2}{9}-3$$

Calculate y from y=5\times \frac{2}{9}-3.

$$y=-\frac{17}{9}$$

Substitute -\frac{17}{9} for y and \frac{2}{9} for z in the equation x=2y-z+4.

$$x=2\left(-\frac{17}{9}\right)-\frac{2}{9}+4$$

Calculate x from x=2\left(-\frac{17}{9}\right)-\frac{2}{9}+4.

$$x=0$$

The system is now solved.

$$x=0 y=-\frac{17}{9} z=\frac{2}{9}$$