Solve {c}{x-y+z=0}{-2x-y+3z=16}{3x+2y-2z=-10}

$$\left. \begin{array} { c } { x - y + z = 0 } \\ { - 2 x - y + 3 z = 16 } \\ { 3 x + 2 y - 2 z = - 10 } \end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=3$$

$$z=5$$

Short Steps Using Substitution

Solve x-y+z=0 for x.

$$x=y-z$$

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

$$-2\left(y-z\right)-y+3z=16 3\left(y-z\right)+2y-2z=-10$$

Solve these equations for y and z respectively.

$$y=\frac{5}{3}z-\frac{16}{3} z=2+y$$

Substitute \frac{5}{3}z-\frac{16}{3} for y in the equation z=2+y.

$$z=2+\frac{5}{3}z-\frac{16}{3}$$

Solve z=2+\frac{5}{3}z-\frac{16}{3} for z.

$$z=5$$

Substitute 5 for z in the equation y=\frac{5}{3}z-\frac{16}{3}.

$$y=\frac{5}{3}\times 5-\frac{16}{3}$$

Calculate y from y=\frac{5}{3}\times 5-\frac{16}{3}.

$$y=3$$

Substitute 3 for y and 5 for z in the equation x=y-z.

$$x=3-5$$

Calculate x from x=3-5.

$$x=-2$$

The system is now solved.

$$x=-2 y=3 z=5$$