Solve {l}{2x-y+z=-3}{3x+2y-z=1}{x-3y+2z=-6}

$$\left. \begin{array} { l } { 2 x - y + z = - 3 } \\ { 3 x + 2 y - z = 1 } \\ { x - 3 y + 2 z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=3$$

$$z=2$$

Short Steps Using Substitution

Solve 2x-y+z=-3 for y.

$$y=2x+z+3$$

Substitute 2x+z+3 for y in the second and third equation.

$$3x+2\left(2x+z+3\right)-z=1 x-3\left(2x+z+3\right)+2z=-6$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{7}z-\frac{5}{7} z=-5x-3$$

Substitute -\frac{1}{7}z-\frac{5}{7} for x in the equation z=-5x-3.

$$z=-5\left(-\frac{1}{7}z-\frac{5}{7}\right)-3$$

Solve z=-5\left(-\frac{1}{7}z-\frac{5}{7}\right)-3 for z.

$$z=2$$

Substitute 2 for z in the equation x=-\frac{1}{7}z-\frac{5}{7}.

$$x=-\frac{1}{7}\times 2-\frac{5}{7}$$

Calculate x from x=-\frac{1}{7}\times 2-\frac{5}{7}.

$$x=-1$$

Substitute -1 for x and 2 for z in the equation y=2x+z+3.

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

Calculate y from y=2\left(-1\right)+2+3.

$$y=3$$

The system is now solved.

$$x=-1 y=3 z=2$$