Solve {l}{x-4y+z=-3}{3x+y-z=18}{x-y-z=7}

$$\left\{ \begin{array} { l } { x - 4 y + z = - 3 } \\ { 3 x + y - z = 18 } \\ { x - y - z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$y=1$$

$$z = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Short Steps Using Substitution

Solve x-4y+z=-3 for x.

$$x=4y-z-3$$

Substitute 4y-z-3 for x in the second and third equation.

$$3\left(4y-z-3\right)+y-z=18 4y-z-3-y-z=7$$

Solve these equations for y and z respectively.

$$y=\frac{4}{13}z+\frac{27}{13} z=\frac{3}{2}y-5$$

Substitute \frac{4}{13}z+\frac{27}{13} for y in the equation z=\frac{3}{2}y-5.

$$z=\frac{3}{2}\left(\frac{4}{13}z+\frac{27}{13}\right)-5$$

Solve z=\frac{3}{2}\left(\frac{4}{13}z+\frac{27}{13}\right)-5 for z.

$$z=-\frac{7}{2}$$

Substitute -\frac{7}{2} for z in the equation y=\frac{4}{13}z+\frac{27}{13}.

$$y=\frac{4}{13}\left(-\frac{7}{2}\right)+\frac{27}{13}$$

Calculate y from y=\frac{4}{13}\left(-\frac{7}{2}\right)+\frac{27}{13}.

$$y=1$$

Substitute 1 for y and -\frac{7}{2} for z in the equation x=4y-z-3.

$$x=4\times 1-\left(-\frac{7}{2}\right)-3$$

Calculate x from x=4\times 1-\left(-\frac{7}{2}\right)-3.

$$x=\frac{9}{2}$$

The system is now solved.

$$x=\frac{9}{2} y=1 z=-\frac{7}{2}$$