Solve {l}{x+y+z=4}{3x-y-2z=0}{x+2y-z=-6}

$$\left\{ \begin{array} { l } { x + y + z = 4 } \\ { 3 x - y - 2 z = 0 } \\ { x + 2 y - z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-2$$

$$z=4$$

Short Steps Using Substitution

Solve x+y+z=4 for x.

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

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

$$3\left(-y-z+4\right)-y-2z=0 -y-z+4+2y-z=-6$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{4}z+3 z=5+\frac{1}{2}y$$

Substitute -\frac{5}{4}z+3 for y in the equation z=5+\frac{1}{2}y.

$$z=5+\frac{1}{2}\left(-\frac{5}{4}z+3\right)$$

Solve z=5+\frac{1}{2}\left(-\frac{5}{4}z+3\right) for z.

$$z=4$$

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

$$y=-\frac{5}{4}\times 4+3$$

Calculate y from y=-\frac{5}{4}\times 4+3.

$$y=-2$$

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

$$x=-\left(-2\right)-4+4$$

Calculate x from x=-\left(-2\right)-4+4.

$$x=2$$

The system is now solved.

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