Solve {l}{x+y-z=6}{x-y+2x=-7}{x+2y+z=4}

$$\left. \begin{array} { l } { x + y - z = 6 } \\ { x - y + 2 x = - 7 } \\ { x + 2 y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=4$$

$$z=-3$$

Short Steps Using Substitution

Solve x+y-z=6 for x.

$$x=-y+z+6$$

Substitute -y+z+6 for x in the second and third equation.

$$-y+z+6-y+2\left(-y+z+6\right)=-7 -y+z+6+2y+z=4$$

Solve these equations for y and z respectively.

$$y=\frac{3}{4}z+\frac{25}{4} z=-\frac{1}{2}y-1$$

Substitute \frac{3}{4}z+\frac{25}{4} for y in the equation z=-\frac{1}{2}y-1.

$$z=-\frac{1}{2}\left(\frac{3}{4}z+\frac{25}{4}\right)-1$$

Solve z=-\frac{1}{2}\left(\frac{3}{4}z+\frac{25}{4}\right)-1 for z.

$$z=-3$$

Substitute -3 for z in the equation y=\frac{3}{4}z+\frac{25}{4}.

$$y=\frac{3}{4}\left(-3\right)+\frac{25}{4}$$

Calculate y from y=\frac{3}{4}\left(-3\right)+\frac{25}{4}.

$$y=4$$

Substitute 4 for y and -3 for z in the equation x=-y+z+6.

$$x=-4-3+6$$

Calculate x from x=-4-3+6.

$$x=-1$$

The system is now solved.

$$x=-1 y=4 z=-3$$