Solve {l}{x-y-z=0}{x+y-3z=4}{2x-5y-7z=21}

$$\left\{ \begin{array} { l } { x - y - z = 0 } \\ { x + y - 3 z = 4 } \\ { 2 x - 5 y - 7 z = 21 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{19}{4} = -4\frac{3}{4} = -4.75$$

$$y = -\frac{11}{8} = -1\frac{3}{8} = -1.375$$

$$z = -\frac{27}{8} = -3\frac{3}{8} = -3.375$$

Short Steps Using Substitution

Solve x-y-z=0 for x.

$$x=y+z$$

Substitute y+z for x in the second and third equation.

$$y+z+y-3z=4 2\left(y+z\right)-5y-7z=21$$

Solve these equations for y and z respectively.

$$y=2+z z=-\frac{21}{5}-\frac{3}{5}y$$

Substitute 2+z for y in the equation z=-\frac{21}{5}-\frac{3}{5}y.

$$z=-\frac{21}{5}-\frac{3}{5}\left(2+z\right)$$

Solve z=-\frac{21}{5}-\frac{3}{5}\left(2+z\right) for z.

$$z=-\frac{27}{8}$$

Substitute -\frac{27}{8} for z in the equation y=2+z.

$$y=2-\frac{27}{8}$$

Calculate y from y=2-\frac{27}{8}.

$$y=-\frac{11}{8}$$

Substitute -\frac{11}{8} for y and -\frac{27}{8} for z in the equation x=y+z.

$$x=-\frac{11}{8}-\frac{27}{8}$$

Calculate x from x=-\frac{11}{8}-\frac{27}{8}.

$$x=-\frac{19}{4}$$

The system is now solved.

$$x=-\frac{19}{4} y=-\frac{11}{8} z=-\frac{27}{8}$$