Solve {l}{x+2y+4z=1}{2x+3y-z=3}{x-3z=2}

$$\left. \begin{array}{l}{ x + 2 y + 4 z = 1 }\\{ 2 x + 3 y - z = 3 }\\{ x - 3 z = 2 }\end{array} \right.$$

Solve for x, y, z

$$x = \frac{19}{11} = 1\frac{8}{11} \approx 1.727272727$$

$$y=-\frac{2}{11}\approx -0.181818182$$

$$z=-\frac{1}{11}\approx -0.090909091$$

Short Steps Using Substitution

Solve x+2y+4z=1 for x.

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

Substitute -2y-4z+1 for x in the second and third equation.

$$2\left(-2y-4z+1\right)+3y-z=3 -2y-4z+1-3z=2$$

Solve these equations for y and z respectively.

$$y=-9z-1 z=-\frac{2}{7}y-\frac{1}{7}$$

Substitute -9z-1 for y in the equation z=-\frac{2}{7}y-\frac{1}{7}.

$$z=-\frac{2}{7}\left(-9z-1\right)-\frac{1}{7}$$

Solve z=-\frac{2}{7}\left(-9z-1\right)-\frac{1}{7} for z.

$$z=-\frac{1}{11}$$

Substitute -\frac{1}{11} for z in the equation y=-9z-1.

$$y=-9\left(-\frac{1}{11}\right)-1$$

Calculate y from y=-9\left(-\frac{1}{11}\right)-1.

$$y=-\frac{2}{11}$$

Substitute -\frac{2}{11} for y and -\frac{1}{11} for z in the equation x=-2y-4z+1.

$$x=-2\left(-\frac{2}{11}\right)-4\left(-\frac{1}{11}\right)+1$$

Calculate x from x=-2\left(-\frac{2}{11}\right)-4\left(-\frac{1}{11}\right)+1.

$$x=\frac{19}{11}$$

The system is now solved.

$$x=\frac{19}{11} y=-\frac{2}{11} z=-\frac{1}{11}$$