Solve {c}{2x+y+3z=11}{x-2y+z=7}{-4x+3y+2z=-11}

$$\left. \begin{array} { c } { 2 x + y + 3 z = 11 } \\ { x - 2 y + z = 7 } \\ { - 4 x + 3 y + 2 z = - 11 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

Solve 2x+y+3z=11 for y.

$$y=-2x-3z+11$$

Substitute -2x-3z+11 for y in the second and third equation.

$$x-2\left(-2x-3z+11\right)+z=7 -4x+3\left(-2x-3z+11\right)+2z=-11$$

Solve these equations for x and z respectively.

$$x=-\frac{7}{5}z+\frac{29}{5} z=\frac{44}{7}-\frac{10}{7}x$$

Substitute -\frac{7}{5}z+\frac{29}{5} for x in the equation z=\frac{44}{7}-\frac{10}{7}x.

$$z=\frac{44}{7}-\frac{10}{7}\left(-\frac{7}{5}z+\frac{29}{5}\right)$$

Solve z=\frac{44}{7}-\frac{10}{7}\left(-\frac{7}{5}z+\frac{29}{5}\right) for z.

$$z=2$$

Substitute 2 for z in the equation x=-\frac{7}{5}z+\frac{29}{5}.

$$x=-\frac{7}{5}\times 2+\frac{29}{5}$$

Calculate x from x=-\frac{7}{5}\times 2+\frac{29}{5}.

$$x=3$$

Substitute 3 for x and 2 for z in the equation y=-2x-3z+11.

$$y=-2\times 3-3\times 2+11$$

Calculate y from y=-2\times 3-3\times 2+11.

$$y=-1$$

The system is now solved.

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