Solve {l}{6x-2y+2z=30}{3x-6y+4z=21}{9x+12y-2z=15}

$$\left. \begin{array} { l } { 6 x - 2 y + 2 z = 30 } \\ { 3 x - 6 y + 4 z = 21 } \\ { 9 x + 12 y - 2 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=-3$$

Short Steps Using Substitution

Solve 6x-2y+2z=30 for x.

$$x=5+\frac{1}{3}y-\frac{1}{3}z$$

Substitute 5+\frac{1}{3}y-\frac{1}{3}z for x in the second and third equation.

$$3\left(5+\frac{1}{3}y-\frac{1}{3}z\right)-6y+4z=21 9\left(5+\frac{1}{3}y-\frac{1}{3}z\right)+12y-2z=15$$

Solve these equations for y and z respectively.

$$y=-\frac{6}{5}+\frac{3}{5}z z=6+3y$$

Substitute -\frac{6}{5}+\frac{3}{5}z for y in the equation z=6+3y.

$$z=6+3\left(-\frac{6}{5}+\frac{3}{5}z\right)$$

Solve z=6+3\left(-\frac{6}{5}+\frac{3}{5}z\right) for z.

$$z=-3$$

Substitute -3 for z in the equation y=-\frac{6}{5}+\frac{3}{5}z.

$$y=-\frac{6}{5}+\frac{3}{5}\left(-3\right)$$

Calculate y from y=-\frac{6}{5}+\frac{3}{5}\left(-3\right).

$$y=-3$$

Substitute -3 for y and -3 for z in the equation x=5+\frac{1}{3}y-\frac{1}{3}z.

$$x=5+\frac{1}{3}\left(-3\right)-\frac{1}{3}\left(-3\right)$$

Calculate x from x=5+\frac{1}{3}\left(-3\right)-\frac{1}{3}\left(-3\right).

$$x=5$$

The system is now solved.

$$x=5 y=-3 z=-3$$