Solve {l}{4x+8y-4z=4}{3x+6y+5z=-13}{-2x+y+12z=-17}

$$\left\{ \begin{array} { l } { 4 x + 8 y - 4 z = 4 } \\ { 3 x + 6 y + 5 z = - 13 } \\ { - 2 x + y + 12 z = - 17 } \end{array} \right.$$

Solve for x, y, z

$$x=-3$$

$$y=1$$

$$z=-2$$

Short Steps Using Substitution

Reorder the equations.

$$-2x+y+12z=-17 3x+6y+5z=-13 4x+8y-4z=4$$

Solve -2x+y+12z=-17 for y.

$$y=2x-12z-17$$

Substitute 2x-12z-17 for y in the second and third equation.

$$3x+6\left(2x-12z-17\right)+5z=-13 4x+8\left(2x-12z-17\right)-4z=4$$

Solve these equations for x and z respectively.

$$x=\frac{67}{15}z+\frac{89}{15} z=-\frac{7}{5}+\frac{1}{5}x$$

Substitute \frac{67}{15}z+\frac{89}{15} for x in the equation z=-\frac{7}{5}+\frac{1}{5}x.

$$z=-\frac{7}{5}+\frac{1}{5}\left(\frac{67}{15}z+\frac{89}{15}\right)$$

Solve z=-\frac{7}{5}+\frac{1}{5}\left(\frac{67}{15}z+\frac{89}{15}\right) for z.

$$z=-2$$

Substitute -2 for z in the equation x=\frac{67}{15}z+\frac{89}{15}.

$$x=\frac{67}{15}\left(-2\right)+\frac{89}{15}$$

Calculate x from x=\frac{67}{15}\left(-2\right)+\frac{89}{15}.

$$x=-3$$

Substitute -3 for x and -2 for z in the equation y=2x-12z-17.

$$y=2\left(-3\right)-12\left(-2\right)-17$$

Calculate y from y=2\left(-3\right)-12\left(-2\right)-17.

$$y=1$$

The system is now solved.

$$x=-3 y=1 z=-2$$