Solve {l}{(-1,5,1)}{x+y+z=5}{x-y+5z=-1}{2x+y+z=4}

$$\left. \begin{array} { l } { ( - 1,5,1 ) } \\ { x + y + z = 5 } \\ { x - y + 5 z = - 1 } \\ { 2 x + y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=5$$

$$z=1$$

Short Steps Using Substitution

Solve x+y+z=5 for x.

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

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

$$-y-z+5-y+5z=-1 2\left(-y-z+5\right)+y+z=4$$

Solve these equations for y and z respectively.

$$y=3+2z z=-y+6$$

Substitute 3+2z for y in the equation z=-y+6.

$$z=-\left(3+2z\right)+6$$

Solve z=-\left(3+2z\right)+6 for z.

$$z=1$$

Substitute 1 for z in the equation y=3+2z.

$$y=3+2\times 1$$

Calculate y from y=3+2\times 1.

$$y=5$$

Substitute 5 for y and 1 for z in the equation x=-y-z+5.

$$x=-5-1+5$$

Calculate x from x=-5-1+5.

$$x=-1$$

The system is now solved.

$$x=-1 y=5 z=1$$