Solve {l}{2x+y-z=1}{x+3y+2z=2}{y+2z=5}

$$\left. \begin{array} { l } { 2 x + y - z = 1 } \\ { x + 3 y + 2 z = 2 } \\ { y + 2 z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{23}{5} = 4\frac{3}{5} = 4.6$$

$$y = -\frac{19}{5} = -3\frac{4}{5} = -3.8$$

$$z = \frac{22}{5} = 4\frac{2}{5} = 4.4$$

Short Steps Using Substitution

Solve 2x+y-z=1 for y.

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

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

$$x+3\left(-2x+z+1\right)+2z=2 -2x+z+1+2z=5$$

Solve these equations for x and z respectively.

$$x=z+\frac{1}{5} z=\frac{4}{3}+\frac{2}{3}x$$

Substitute z+\frac{1}{5} for x in the equation z=\frac{4}{3}+\frac{2}{3}x.

$$z=\frac{4}{3}+\frac{2}{3}\left(z+\frac{1}{5}\right)$$

Solve z=\frac{4}{3}+\frac{2}{3}\left(z+\frac{1}{5}\right) for z.

$$z=\frac{22}{5}$$

Substitute \frac{22}{5} for z in the equation x=z+\frac{1}{5}.

$$x=\frac{22}{5}+\frac{1}{5}$$

Calculate x from x=\frac{22}{5}+\frac{1}{5}.

$$x=\frac{23}{5}$$

Substitute \frac{23}{5} for x and \frac{22}{5} for z in the equation y=-2x+z+1.

$$y=-2\times \frac{23}{5}+\frac{22}{5}+1$$

Calculate y from y=-2\times \frac{23}{5}+\frac{22}{5}+1.

$$y=-\frac{19}{5}$$

The system is now solved.

$$x=\frac{23}{5} y=-\frac{19}{5} z=\frac{22}{5}$$