Solve {l}{3x-y+2z=5}{2y-z=-3}{x-2y+3z=8}

$$\left\{ \begin{array} { l } { 3 x - y + 2 z = 5 } \\ { 2 y - z = - 3 } \\ { x - 2 y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{1}{9}\approx -0.111111111$$

$$y=-\frac{2}{9}\approx -0.222222222$$

$$z = \frac{23}{9} = 2\frac{5}{9} \approx 2.555555556$$

Short Steps Using Substitution

Solve 3x-y+2z=5 for y.

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

Substitute 3x+2z-5 for y in the second and third equation.

$$2\left(3x+2z-5\right)-z=-3 x-2\left(3x+2z-5\right)+3z=8$$

Solve these equations for x and z respectively.

$$x=\frac{7}{6}-\frac{1}{2}z z=-5x+2$$

Substitute \frac{7}{6}-\frac{1}{2}z for x in the equation z=-5x+2.

$$z=-5\left(\frac{7}{6}-\frac{1}{2}z\right)+2$$

Solve z=-5\left(\frac{7}{6}-\frac{1}{2}z\right)+2 for z.

$$z=\frac{23}{9}$$

Substitute \frac{23}{9} for z in the equation x=\frac{7}{6}-\frac{1}{2}z.

$$x=\frac{7}{6}-\frac{1}{2}\times \frac{23}{9}$$

Calculate x from x=\frac{7}{6}-\frac{1}{2}\times \frac{23}{9}.

$$x=-\frac{1}{9}$$

Substitute -\frac{1}{9} for x and \frac{23}{9} for z in the equation y=3x+2z-5.

$$y=3\left(-\frac{1}{9}\right)+2\times \frac{23}{9}-5$$

Calculate y from y=3\left(-\frac{1}{9}\right)+2\times \frac{23}{9}-5.

$$y=-\frac{2}{9}$$

The system is now solved.

$$x=-\frac{1}{9} y=-\frac{2}{9} z=\frac{23}{9}$$