Solve {l}{2x+3y-5z=71}{4x+y-2z=-8}{x-2y+z=-2}

$$\left\{ \begin{array} { l } { 2 x + 3 y - 5 z = 71 } \\ { 4 x + y - 2 z = - 8 } \\ { x - 2 y + z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{89}{7} = -12\frac{5}{7} \approx -12.714285714$$

$$y = -\frac{150}{7} = -21\frac{3}{7} \approx -21.428571429$$

$$z = -\frac{225}{7} = -32\frac{1}{7} \approx -32.142857143$$

Short Steps Using Substitution

Reorder the equations.

$$4x+y-2z=-8 2x+3y-5z=71 x-2y+z=-2$$

Solve 4x+y-2z=-8 for y.

$$y=-8-4x+2z$$

Substitute -8-4x+2z for y in the second and third equation.

$$2x+3\left(-8-4x+2z\right)-5z=71 x-2\left(-8-4x+2z\right)+z=-2$$

Solve these equations for x and z respectively.

$$x=-\frac{19}{2}+\frac{1}{10}z z=6+3x$$

Substitute -\frac{19}{2}+\frac{1}{10}z for x in the equation z=6+3x.

$$z=6+3\left(-\frac{19}{2}+\frac{1}{10}z\right)$$

Solve z=6+3\left(-\frac{19}{2}+\frac{1}{10}z\right) for z.

$$z=-\frac{225}{7}$$

Substitute -\frac{225}{7} for z in the equation x=-\frac{19}{2}+\frac{1}{10}z.

$$x=-\frac{19}{2}+\frac{1}{10}\left(-\frac{225}{7}\right)$$

Calculate x from x=-\frac{19}{2}+\frac{1}{10}\left(-\frac{225}{7}\right).

$$x=-\frac{89}{7}$$

Substitute -\frac{89}{7} for x and -\frac{225}{7} for z in the equation y=-8-4x+2z.

$$y=-8-4\left(-\frac{89}{7}\right)+2\left(-\frac{225}{7}\right)$$

Calculate y from y=-8-4\left(-\frac{89}{7}\right)+2\left(-\frac{225}{7}\right).

$$y=-\frac{150}{7}$$

The system is now solved.

$$x=-\frac{89}{7} y=-\frac{150}{7} z=-\frac{225}{7}$$