Solve {c}{2x-y+z=2}{3x+y-2z=9}{-x+2y+5z=-5}

$$\left\{ \begin{array} { c } { 2 x - y + z = 2 } \\ { 3 x + y - 2 z = 9 } \\ { - x + 2 y + 5 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=1$$

$$z=-1$$

Short Steps Using Substitution

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

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

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

$$3x+2x+z-2-2z=9 -x+2\left(2x+z-2\right)+5z=-5$$

Solve these equations for x and z respectively.

$$x=\frac{11}{5}+\frac{1}{5}z z=-\frac{3}{7}x-\frac{1}{7}$$

Substitute \frac{11}{5}+\frac{1}{5}z for x in the equation z=-\frac{3}{7}x-\frac{1}{7}.

$$z=-\frac{3}{7}\left(\frac{11}{5}+\frac{1}{5}z\right)-\frac{1}{7}$$

Solve z=-\frac{3}{7}\left(\frac{11}{5}+\frac{1}{5}z\right)-\frac{1}{7} for z.

$$z=-1$$

Substitute -1 for z in the equation x=\frac{11}{5}+\frac{1}{5}z.

$$x=\frac{11}{5}+\frac{1}{5}\left(-1\right)$$

Calculate x from x=\frac{11}{5}+\frac{1}{5}\left(-1\right).

$$x=2$$

Substitute 2 for x and -1 for z in the equation y=2x+z-2.

$$y=2\times 2-1-2$$

Calculate y from y=2\times 2-1-2.

$$y=1$$

The system is now solved.

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