Solve {l}{x+y+z=6}{x-y+2z=5}{x-y-3z=-10}

$$\left\{ \begin{array} { l } { x + y + z = 6 } \\ { x - y + 2 z = 5 } \\ { x - y - 3 z = - 10 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

$$x=-y-z+6$$

Substitute -y-z+6 for x in the second and third equation.

$$-y-z+6-y+2z=5 -y-z+6-y-3z=-10$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=3$$

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

$$y=\frac{1}{2}+\frac{1}{2}\times 3$$

Calculate y from y=\frac{1}{2}+\frac{1}{2}\times 3.

$$y=2$$

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

$$x=-2-3+6$$

Calculate x from x=-2-3+6.

$$x=1$$

The system is now solved.

$$x=1 y=2 z=3$$