Solve {c}{2x-y+z=7}{x+2y-z=6}{-x+y+2z=11}

$$\left\{ \begin{array} { c } { 2 x - y + z = 7 } \\ { x + 2 y - z = 6 } \\ { - x + y + 2 z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=4$$

$$z=5$$

Short Steps Using Substitution

Solve 2x-y+z=7 for y.

$$y=2x+z-7$$

Substitute 2x+z-7 for y in the second and third equation.

$$x+2\left(2x+z-7\right)-z=6 -x+2x+z-7+2z=11$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{5}z+4 z=-\frac{1}{3}x+6$$

Substitute -\frac{1}{5}z+4 for x in the equation z=-\frac{1}{3}x+6.

$$z=-\frac{1}{3}\left(-\frac{1}{5}z+4\right)+6$$

Solve z=-\frac{1}{3}\left(-\frac{1}{5}z+4\right)+6 for z.

$$z=5$$

Substitute 5 for z in the equation x=-\frac{1}{5}z+4.

$$x=-\frac{1}{5}\times 5+4$$

Calculate x from x=-\frac{1}{5}\times 5+4.

$$x=3$$

Substitute 3 for x and 5 for z in the equation y=2x+z-7.

$$y=2\times 3+5-7$$

Calculate y from y=2\times 3+5-7.

$$y=4$$

The system is now solved.

$$x=3 y=4 z=5$$