Solve {l}{x-4y+z=5}{4x+2y+z=2}{-4x+y-3z=-8}

$$\left\{ \begin{array}{l}{ x - 4 y + z = 5 }\\{ 4 x + 2 y + z = 2 }\\{ - 4 x + y - 3 z = - 8 }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=-\frac{2}{3}\approx -0.666666667$$

$$z=2$$

Short Steps Using Substitution

Solve x-4y+z=5 for x.

$$x=4y-z+5$$

Substitute 4y-z+5 for x in the second and third equation.

$$4\left(4y-z+5\right)+2y+z=2 -4\left(4y-z+5\right)+y-3z=-8$$

Solve these equations for y and z respectively.

$$y=-1+\frac{1}{6}z z=12+15y$$

Substitute -1+\frac{1}{6}z for y in the equation z=12+15y.

$$z=12+15\left(-1+\frac{1}{6}z\right)$$

Solve z=12+15\left(-1+\frac{1}{6}z\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=-1+\frac{1}{6}z.

$$y=-1+\frac{1}{6}\times 2$$

Calculate y from y=-1+\frac{1}{6}\times 2.

$$y=-\frac{2}{3}$$

Substitute -\frac{2}{3} for y and 2 for z in the equation x=4y-z+5.

$$x=4\left(-\frac{2}{3}\right)-2+5$$

Calculate x from x=4\left(-\frac{2}{3}\right)-2+5.

$$x=\frac{1}{3}$$

The system is now solved.

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