Solve {l}{x+2y-2z=3}{2x-y+3z=-5}{4x-3y+z=7}

$$\left. \begin{array} { l } { x + 2 y - 2 z = 3 } \\ { 2 x - y + 3 z = - 5 } \\ { 4 x - 3 y + z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-2$$

$$z=-3$$

Short Steps Using Substitution

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

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

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

$$2\left(-2y+2z+3\right)-y+3z=-5 4\left(-2y+2z+3\right)-3y+z=7$$

Solve these equations for y and z respectively.

$$y=\frac{7}{5}z+\frac{11}{5} z=\frac{11}{9}y-\frac{5}{9}$$

Substitute \frac{7}{5}z+\frac{11}{5} for y in the equation z=\frac{11}{9}y-\frac{5}{9}.

$$z=\frac{11}{9}\left(\frac{7}{5}z+\frac{11}{5}\right)-\frac{5}{9}$$

Solve z=\frac{11}{9}\left(\frac{7}{5}z+\frac{11}{5}\right)-\frac{5}{9} for z.

$$z=-3$$

Substitute -3 for z in the equation y=\frac{7}{5}z+\frac{11}{5}.

$$y=\frac{7}{5}\left(-3\right)+\frac{11}{5}$$

Calculate y from y=\frac{7}{5}\left(-3\right)+\frac{11}{5}.

$$y=-2$$

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

$$x=-2\left(-2\right)+2\left(-3\right)+3$$

Calculate x from x=-2\left(-2\right)+2\left(-3\right)+3.

$$x=1$$

The system is now solved.

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