Solve {r}{2x-5y-2z=3}{x+2y+1z=5}{-3x-y+2z=6}

$$\left. \begin{array} { r } { 2 x - 5 y - 2 z = 3 } \\ { x + 2 y + 1 z = 5 } \\ { - 3 x - y + 2 z = 6 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{69}{25} = 2\frac{19}{25} = 2.76$$

$$y = -\frac{49}{25} = -1\frac{24}{25} = -1.96$$

$$z = \frac{154}{25} = 6\frac{4}{25} = 6.16$$

Short Steps Using Substitution

Reorder the equations.

$$x+2y+1z=5 2x-5y-2z=3 -3x-y+2z=6$$

Solve x+2y+1z=5 for x.

$$x=-2y-z+5$$

Substitute -2y-z+5 for x in the second and third equation.

$$2\left(-2y-z+5\right)-5y-2z=3 -3\left(-2y-z+5\right)-y+2z=6$$

Solve these equations for y and z respectively.

$$y=-\frac{4}{9}z+\frac{7}{9} z=-y+\frac{21}{5}$$

Substitute -\frac{4}{9}z+\frac{7}{9} for y in the equation z=-y+\frac{21}{5}.

$$z=-\left(-\frac{4}{9}z+\frac{7}{9}\right)+\frac{21}{5}$$

Solve z=-\left(-\frac{4}{9}z+\frac{7}{9}\right)+\frac{21}{5} for z.

$$z=\frac{154}{25}$$

Substitute \frac{154}{25} for z in the equation y=-\frac{4}{9}z+\frac{7}{9}.

$$y=-\frac{4}{9}\times \frac{154}{25}+\frac{7}{9}$$

Calculate y from y=-\frac{4}{9}\times \frac{154}{25}+\frac{7}{9}.

$$y=-\frac{49}{25}$$

Substitute -\frac{49}{25} for y and \frac{154}{25} for z in the equation x=-2y-z+5.

$$x=-2\left(-\frac{49}{25}\right)-\frac{154}{25}+5$$

Calculate x from x=-2\left(-\frac{49}{25}\right)-\frac{154}{25}+5.

$$x=\frac{69}{25}$$

The system is now solved.

$$x=\frac{69}{25} y=-\frac{49}{25} z=\frac{154}{25}$$