Solve {l}{x-y-z=1}{x-2y+3z=4}{3x-2y+4z=0}

$$\left. \begin{array} { l } { x - y - z = 1 } \\ { x - 2 y + 3 z = 4 } \\ { 3 x - 2 y + 4 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=-3$$

$$z=0$$

Short Steps Using Substitution

Solve x-y-z=1 for x.

$$x=y+z+1$$

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

$$y+z+1-2y+3z=4 3\left(y+z+1\right)-2y+4z=0$$

Solve these equations for y and z respectively.

$$y=4z-3 z=-\frac{3}{7}-\frac{1}{7}y$$

Substitute 4z-3 for y in the equation z=-\frac{3}{7}-\frac{1}{7}y.

$$z=-\frac{3}{7}-\frac{1}{7}\left(4z-3\right)$$

Solve z=-\frac{3}{7}-\frac{1}{7}\left(4z-3\right) for z.

$$z=0$$

Substitute 0 for z in the equation y=4z-3.

$$y=4\times 0-3$$

Calculate y from y=4\times 0-3.

$$y=-3$$

Substitute -3 for y and 0 for z in the equation x=y+z+1.

$$x=-3+0+1$$

Calculate x from x=-3+0+1.

$$x=-2$$

The system is now solved.

$$x=-2 y=-3 z=0$$