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

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

Solve for x, y, z

$$x = \frac{19}{10} = 1\frac{9}{10} = 1.9$$

$$y=\frac{3}{10}=0.3$$

$$z = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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.

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

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

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

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

$$z=-\frac{3}{2}$$

Substitute -\frac{3}{2} for z in the equation x=\frac{11}{5}+\frac{1}{5}z.

$$x=\frac{11}{5}+\frac{1}{5}\left(-\frac{3}{2}\right)$$

Calculate x from x=\frac{11}{5}+\frac{1}{5}\left(-\frac{3}{2}\right).

$$x=\frac{19}{10}$$

Substitute \frac{19}{10} for x and -\frac{3}{2} for z in the equation y=2x+z-2.

$$y=2\times \frac{19}{10}-\frac{3}{2}-2$$

Calculate y from y=2\times \frac{19}{10}-\frac{3}{2}-2.

$$y=\frac{3}{10}$$

The system is now solved.

$$x=\frac{19}{10} y=\frac{3}{10} z=-\frac{3}{2}$$