Solve {r}{2x+5y+2z=-38}{3x-2y+4z=17}{-6x+y-7z=-12}

$$\left. \begin{array} { r } { 2 x + 5 y + 2 z = - 38 } \\ { 3 x - 2 y + 4 z = 17 } \\ { - 6 x + y - 7 z = - 12 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-8$$

$$z=-2$$

Short Steps Using Substitution

Reorder the equations.

$$-6x+y-7z=-12 3x-2y+4z=17 2x+5y+2z=-38$$

Solve -6x+y-7z=-12 for y.

$$y=6x+7z-12$$

Substitute 6x+7z-12 for y in the second and third equation.

$$3x-2\left(6x+7z-12\right)+4z=17 2x+5\left(6x+7z-12\right)+2z=-38$$

Solve these equations for x and z respectively.

$$x=-\frac{10}{9}z+\frac{7}{9} z=\frac{22}{37}-\frac{32}{37}x$$

Substitute -\frac{10}{9}z+\frac{7}{9} for x in the equation z=\frac{22}{37}-\frac{32}{37}x.

$$z=\frac{22}{37}-\frac{32}{37}\left(-\frac{10}{9}z+\frac{7}{9}\right)$$

Solve z=\frac{22}{37}-\frac{32}{37}\left(-\frac{10}{9}z+\frac{7}{9}\right) for z.

$$z=-2$$

Substitute -2 for z in the equation x=-\frac{10}{9}z+\frac{7}{9}.

$$x=-\frac{10}{9}\left(-2\right)+\frac{7}{9}$$

Calculate x from x=-\frac{10}{9}\left(-2\right)+\frac{7}{9}.

$$x=3$$

Substitute 3 for x and -2 for z in the equation y=6x+7z-12.

$$y=6\times 3+7\left(-2\right)-12$$

Calculate y from y=6\times 3+7\left(-2\right)-12.

$$y=-8$$

The system is now solved.

$$x=3 y=-8 z=-2$$