Solve {l}{4x+2y-2z=-8}{-6x+4y+2z=0}{2x+2y+12z=60}

$$\left\{ \begin{array} { l } { 4 x + 2 y - 2 z = - 8 } \\ { - 6 x + 4 y + 2 z = 0 } \\ { 2 x + 2 y + 12 z = 60 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-1$$

$$z=5$$

Short Steps Using Substitution

Solve 4x+2y-2z=-8 for x.

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

Substitute -2-\frac{1}{2}y+\frac{1}{2}z for x in the second and third equation.

$$-6\left(-2-\frac{1}{2}y+\frac{1}{2}z\right)+4y+2z=0 2\left(-2-\frac{1}{2}y+\frac{1}{2}z\right)+2y+12z=60$$

Solve these equations for y and z respectively.

$$y=-\frac{12}{7}+\frac{1}{7}z z=\frac{64}{13}-\frac{1}{13}y$$

Substitute -\frac{12}{7}+\frac{1}{7}z for y in the equation z=\frac{64}{13}-\frac{1}{13}y.

$$z=\frac{64}{13}-\frac{1}{13}\left(-\frac{12}{7}+\frac{1}{7}z\right)$$

Solve z=\frac{64}{13}-\frac{1}{13}\left(-\frac{12}{7}+\frac{1}{7}z\right) for z.

$$z=5$$

Substitute 5 for z in the equation y=-\frac{12}{7}+\frac{1}{7}z.

$$y=-\frac{12}{7}+\frac{1}{7}\times 5$$

Calculate y from y=-\frac{12}{7}+\frac{1}{7}\times 5.

$$y=-1$$

Substitute -1 for y and 5 for z in the equation x=-2-\frac{1}{2}y+\frac{1}{2}z.

$$x=-2-\frac{1}{2}\left(-1\right)+\frac{1}{2}\times 5$$

Calculate x from x=-2-\frac{1}{2}\left(-1\right)+\frac{1}{2}\times 5.

$$x=1$$

The system is now solved.

$$x=1 y=-1 z=5$$