Solve {l}{x+2y+5z=-3}{2x-2y-4z=1}{4x+4y+z=-5}

$$\left. \begin{array} { l } { x + 2 y + 5 z = - 3 } \\ { 2 x - 2 y - 4 z = 1 } \\ { 4 x + 4 y + z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{17}{29}\approx -0.586206897$$

$$y=-\frac{35}{58}\approx -0.603448276$$

$$z=-\frac{7}{29}\approx -0.24137931$$

Short Steps Using Substitution

Solve x+2y+5z=-3 for x.

$$x=-2y-5z-3$$

Substitute -2y-5z-3 for x in the second and third equation.

$$2\left(-2y-5z-3\right)-2y-4z=1 4\left(-2y-5z-3\right)+4y+z=-5$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{6}-\frac{7}{3}z z=-\frac{4}{19}y-\frac{7}{19}$$

Substitute -\frac{7}{6}-\frac{7}{3}z for y in the equation z=-\frac{4}{19}y-\frac{7}{19}.

$$z=-\frac{4}{19}\left(-\frac{7}{6}-\frac{7}{3}z\right)-\frac{7}{19}$$

Solve z=-\frac{4}{19}\left(-\frac{7}{6}-\frac{7}{3}z\right)-\frac{7}{19} for z.

$$z=-\frac{7}{29}$$

Substitute -\frac{7}{29} for z in the equation y=-\frac{7}{6}-\frac{7}{3}z.

$$y=-\frac{7}{6}-\frac{7}{3}\left(-\frac{7}{29}\right)$$

Calculate y from y=-\frac{7}{6}-\frac{7}{3}\left(-\frac{7}{29}\right).

$$y=-\frac{35}{58}$$

Substitute -\frac{35}{58} for y and -\frac{7}{29} for z in the equation x=-2y-5z-3.

$$x=-2\left(-\frac{35}{58}\right)-5\left(-\frac{7}{29}\right)-3$$

Calculate x from x=-2\left(-\frac{35}{58}\right)-5\left(-\frac{7}{29}\right)-3.

$$x=-\frac{17}{29}$$

The system is now solved.

$$x=-\frac{17}{29} y=-\frac{35}{58} z=-\frac{7}{29}$$