Solve {r}{x+5y+z=5}{5x+y-z=1}{x-4y-z=-3}

$$\left. \begin{array} { r } { x + 5 y + z = 5 } \\ { 5 x + y - z = 1 } \\ { x - 4 y - z = - 3 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=0$$

$$z=4$$

Short Steps Using Substitution

Solve x+5y+z=5 for x.

$$x=-5y-z+5$$

Substitute -5y-z+5 for x in the second and third equation.

$$5\left(-5y-z+5\right)+y-z=1 -5y-z+5-4y-z=-3$$

Solve these equations for y and z respectively.

$$y=1-\frac{1}{4}z z=-\frac{9}{2}y+4$$

Substitute 1-\frac{1}{4}z for y in the equation z=-\frac{9}{2}y+4.

$$z=-\frac{9}{2}\left(1-\frac{1}{4}z\right)+4$$

Solve z=-\frac{9}{2}\left(1-\frac{1}{4}z\right)+4 for z.

$$z=4$$

Substitute 4 for z in the equation y=1-\frac{1}{4}z.

$$y=1-\frac{1}{4}\times 4$$

Calculate y from y=1-\frac{1}{4}\times 4.

$$y=0$$

Substitute 0 for y and 4 for z in the equation x=-5y-z+5.

$$x=-5\times 0-4+5$$

Calculate x from x=-5\times 0-4+5.

$$x=1$$

The system is now solved.

$$x=1 y=0 z=4$$