Solve {l}{x-2y+z=9}{2x+3y-2z=-12}{x+y-3z=-13}

$$\left\{ \begin{array} { l } { x - 2 y + z = 9 } \\ { 2 x + 3 y - 2 z = - 12 } \\ { x + y - 3 z = - 13 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-2$$

$$z=4$$

Short Steps Using Substitution

Solve x-2y+z=9 for x.

$$x=2y-z+9$$

Substitute 2y-z+9 for x in the second and third equation.

$$2\left(2y-z+9\right)+3y-2z=-12 2y-z+9+y-3z=-13$$

Solve these equations for y and z respectively.

$$y=-\frac{30}{7}+\frac{4}{7}z z=\frac{3}{4}y+\frac{11}{2}$$

Substitute -\frac{30}{7}+\frac{4}{7}z for y in the equation z=\frac{3}{4}y+\frac{11}{2}.

$$z=\frac{3}{4}\left(-\frac{30}{7}+\frac{4}{7}z\right)+\frac{11}{2}$$

Solve z=\frac{3}{4}\left(-\frac{30}{7}+\frac{4}{7}z\right)+\frac{11}{2} for z.

$$z=4$$

Substitute 4 for z in the equation y=-\frac{30}{7}+\frac{4}{7}z.

$$y=-\frac{30}{7}+\frac{4}{7}\times 4$$

Calculate y from y=-\frac{30}{7}+\frac{4}{7}\times 4.

$$y=-2$$

Substitute -2 for y and 4 for z in the equation x=2y-z+9.

$$x=2\left(-2\right)-4+9$$

Calculate x from x=2\left(-2\right)-4+9.

$$x=1$$

The system is now solved.

$$x=1 y=-2 z=4$$