Solve x-2y+3z=2;2x-3z=3,x+y+z=0

$$x - 2 y + 3 z = 2 ; 2 x - 3 z = 3 , x + y + z = 0$$

Solve for x, y, z

$$x = \frac{21}{19} = 1\frac{2}{19} \approx 1.105263158$$

$$y=-\frac{16}{19}\approx -0.842105263$$

$$z=-\frac{5}{19}\approx -0.263157895$$

Short Steps Using Substitution

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

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

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

$$2\left(2y-3z+2\right)-3z=3 2y-3z+2+y+z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{4}+\frac{9}{4}z z=\frac{3}{2}y+1$$

Substitute -\frac{1}{4}+\frac{9}{4}z for y in the equation z=\frac{3}{2}y+1.

$$z=\frac{3}{2}\left(-\frac{1}{4}+\frac{9}{4}z\right)+1$$

Solve z=\frac{3}{2}\left(-\frac{1}{4}+\frac{9}{4}z\right)+1 for z.

$$z=-\frac{5}{19}$$

Substitute -\frac{5}{19} for z in the equation y=-\frac{1}{4}+\frac{9}{4}z.

$$y=-\frac{1}{4}+\frac{9}{4}\left(-\frac{5}{19}\right)$$

Calculate y from y=-\frac{1}{4}+\frac{9}{4}\left(-\frac{5}{19}\right).

$$y=-\frac{16}{19}$$

Substitute -\frac{16}{19} for y and -\frac{5}{19} for z in the equation x=2y-3z+2.

$$x=2\left(-\frac{16}{19}\right)-3\left(-\frac{5}{19}\right)+2$$

Calculate x from x=2\left(-\frac{16}{19}\right)-3\left(-\frac{5}{19}\right)+2.

$$x=\frac{21}{19}$$

The system is now solved.

$$x=\frac{21}{19} y=-\frac{16}{19} z=-\frac{5}{19}$$