Solve {l}{x-4y+z=-30}{2x+y-z=18(2)}{x-y-z=7}

$$\left\{ \begin{array} { l } { x - 4 y + z = - 30 } \\ { 2 x + y - z = 18 ( 2 ) } \\ { x - y - z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=11$$

$$y=9$$

$$z=-5$$

Short Steps Using Substitution

Solve x-4y+z=-30 for x.

$$x=4y-z-30$$

Substitute 4y-z-30 for x in the second and third equation.

$$2\left(4y-z-30\right)+y-z=18\times 2 4y-z-30-y-z=7$$

Solve these equations for y and z respectively.

$$y=\frac{32}{3}+\frac{1}{3}z z=\frac{3}{2}y-\frac{37}{2}$$

Substitute \frac{32}{3}+\frac{1}{3}z for y in the equation z=\frac{3}{2}y-\frac{37}{2}.

$$z=\frac{3}{2}\left(\frac{32}{3}+\frac{1}{3}z\right)-\frac{37}{2}$$

Solve z=\frac{3}{2}\left(\frac{32}{3}+\frac{1}{3}z\right)-\frac{37}{2} for z.

$$z=-5$$

Substitute -5 for z in the equation y=\frac{32}{3}+\frac{1}{3}z.

$$y=\frac{32}{3}+\frac{1}{3}\left(-5\right)$$

Calculate y from y=\frac{32}{3}+\frac{1}{3}\left(-5\right).

$$y=9$$

Substitute 9 for y and -5 for z in the equation x=4y-z-30.

$$x=4\times 9-\left(-5\right)-30$$

Calculate x from x=4\times 9-\left(-5\right)-30.

$$x=11$$

The system is now solved.

$$x=11 y=9 z=-5$$