Solve {l}{3x-2y+z=6}{6x+y-2z=-2}{6x+2y+5z=3}

$$\left. \begin{array} { l } { 3 x - 2 y + z = 6 } \\ { 6 x + y - 2 z = - 2 } \\ { 6 x + 2 y + 5 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=-2$$

$$z=1$$

Short Steps Using Substitution

Solve 3x-2y+z=6 for z.

$$z=-3x+2y+6$$

Substitute -3x+2y+6 for z in the second and third equation.

$$6x+y-2\left(-3x+2y+6\right)=-2 6x+2y+5\left(-3x+2y+6\right)=3$$

Solve these equations for y and x respectively.

$$y=-\frac{10}{3}+4x x=3+\frac{4}{3}y$$

Substitute -\frac{10}{3}+4x for y in the equation x=3+\frac{4}{3}y.

$$x=3+\frac{4}{3}\left(-\frac{10}{3}+4x\right)$$

Solve x=3+\frac{4}{3}\left(-\frac{10}{3}+4x\right) for x.

$$x=\frac{1}{3}$$

Substitute \frac{1}{3} for x in the equation y=-\frac{10}{3}+4x.

$$y=-\frac{10}{3}+4\times \frac{1}{3}$$

Calculate y from y=-\frac{10}{3}+4\times \frac{1}{3}.

$$y=-2$$

Substitute -2 for y and \frac{1}{3} for x in the equation z=-3x+2y+6.

$$z=-3\times \frac{1}{3}+2\left(-2\right)+6$$

Calculate z from z=-3\times \frac{1}{3}+2\left(-2\right)+6.

$$z=1$$

The system is now solved.

$$x=\frac{1}{3} y=-2 z=1$$