Solve {l}{-3x+2y-z=-2}{5x-3y+2z=10}{+2x-2y-3z=-9}

$$\left\{ \begin{array} { l } { - 3 x + 2 y - z = - 2 } \\ { 5 x - 3 y + 2 z = 10 } \\ { + 2 x - 2 y - 3 z = - 9 } \end{array} \right.$$

Solve for x, y, z

$$x=15$$

$$y=21$$

$$z=-1$$

Short Steps Using Substitution

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

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

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

$$5x-3y+2\left(-3x+2y+2\right)=10 2x-2y-3\left(-3x+2y+2\right)=-9$$

Solve these equations for y and x respectively.

$$y=x+6 x=\frac{8}{11}y-\frac{3}{11}$$

Substitute x+6 for y in the equation x=\frac{8}{11}y-\frac{3}{11}.

$$x=\frac{8}{11}\left(x+6\right)-\frac{3}{11}$$

Solve x=\frac{8}{11}\left(x+6\right)-\frac{3}{11} for x.

$$x=15$$

Substitute 15 for x in the equation y=x+6.

$$y=15+6$$

Calculate y from y=15+6.

$$y=21$$

Substitute 21 for y and 15 for x in the equation z=-3x+2y+2.

$$z=-3\times 15+2\times 21+2$$

Calculate z from z=-3\times 15+2\times 21+2.

$$z=-1$$

The system is now solved.

$$x=15 y=21 z=-1$$