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

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

Solve for x, y, z

$$x=\frac{2}{25}=0.08$$

$$y = \frac{58}{25} = 2\frac{8}{25} = 2.32$$

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

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.

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

Solve these equations for y and x respectively.

$$y=\frac{22}{9}-\frac{14}{9}x x=-y+\frac{12}{5}$$

Substitute \frac{22}{9}-\frac{14}{9}x for y in the equation x=-y+\frac{12}{5}.

$$x=-\left(\frac{22}{9}-\frac{14}{9}x\right)+\frac{12}{5}$$

Solve x=-\left(\frac{22}{9}-\frac{14}{9}x\right)+\frac{12}{5} for x.

$$x=\frac{2}{25}$$

Substitute \frac{2}{25} for x in the equation y=\frac{22}{9}-\frac{14}{9}x.

$$y=\frac{22}{9}-\frac{14}{9}\times \frac{2}{25}$$

Calculate y from y=\frac{22}{9}-\frac{14}{9}\times \frac{2}{25}.

$$y=\frac{58}{25}$$

Substitute \frac{58}{25} for y and \frac{2}{25} for x in the equation z=-3x-2y+5.

$$z=-3\times \frac{2}{25}-2\times \frac{58}{25}+5$$

Calculate z from z=-3\times \frac{2}{25}-2\times \frac{58}{25}+5.

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

The system is now solved.

$$x=\frac{2}{25} y=\frac{58}{25} z=\frac{3}{25}$$