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

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

Solve for x, y, z

$$x = -\frac{52}{25} = -2\frac{2}{25} = -2.08$$

$$y = \frac{142}{25} = 5\frac{17}{25} = 5.68$$

$$z=-\frac{3}{25}=-0.12$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and x respectively.

$$y=\frac{22}{9}-\frac{14}{9}x x=-y+\frac{18}{5}$$

Substitute \frac{22}{9}-\frac{14}{9}x for y in the equation x=-y+\frac{18}{5}.

$$x=-\left(\frac{22}{9}-\frac{14}{9}x\right)+\frac{18}{5}$$

Solve x=-\left(\frac{22}{9}-\frac{14}{9}x\right)+\frac{18}{5} for x.

$$x=-\frac{52}{25}$$

Substitute -\frac{52}{25} for x in the equation y=\frac{22}{9}-\frac{14}{9}x.

$$y=\frac{22}{9}-\frac{14}{9}\left(-\frac{52}{25}\right)$$

Calculate y from y=\frac{22}{9}-\frac{14}{9}\left(-\frac{52}{25}\right).

$$y=\frac{142}{25}$$

Substitute \frac{142}{25} for y and -\frac{52}{25} for x in the equation z=-3x-2y+5.

$$z=-3\left(-\frac{52}{25}\right)-2\times \frac{142}{25}+5$$

Calculate z from z=-3\left(-\frac{52}{25}\right)-2\times \frac{142}{25}+5.

$$z=-\frac{3}{25}$$

The system is now solved.

$$x=-\frac{52}{25} y=\frac{142}{25} z=-\frac{3}{25}$$