Solve {l}{x+y+z=-1}{2x-2y+z=-10}{3x+y-z=-7}

$$\left. \begin{array} { l } { x + y + z = - 1 } \\ { 2 x - 2 y + z = - 10 } \\ { 3 x + y - z = - 7 } \end{array} \right.$$

Solve for x, y, z

$$x=-3$$

$$y=2$$

$$z=0$$

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.

$$2\left(-y-z-1\right)-2y+z=-10 3\left(-y-z-1\right)+y-z=-7$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=0$$

Substitute 0 for z in the equation y=-\frac{1}{4}z+2.

$$y=-\frac{1}{4}\times 0+2$$

Calculate y from y=-\frac{1}{4}\times 0+2.

$$y=2$$

Substitute 2 for y and 0 for z in the equation x=-y-z-1.

$$x=-2-0-1$$

Calculate x from x=-2-0-1.

$$x=-3$$

The system is now solved.

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