Solve {l}{x-5y+12z=-66}{x+3y+3z=19}{x+5y-4z=49}

$$\left. \begin{array} { l } { x - 5 y + 12 z = - 66 } \\ { x + 3 y + 3 z = 19 } \\ { x + 5 y - 4 z = 49 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{43}{38} = -1\frac{5}{38} \approx -1.131578947$$

$$y = \frac{325}{38} = 8\frac{21}{38} \approx 8.552631579$$

$$z = -\frac{35}{19} = -1\frac{16}{19} \approx -1.842105263$$

Short Steps Using Substitution

Solve x-5y+12z=-66 for x.

$$x=5y-12z-66$$

Substitute 5y-12z-66 for x in the second and third equation.

$$5y-12z-66+3y+3z=19 5y-12z-66+5y-4z=49$$

Solve these equations for y and z respectively.

$$y=\frac{9}{8}z+\frac{85}{8} z=-\frac{115}{16}+\frac{5}{8}y$$

Substitute \frac{9}{8}z+\frac{85}{8} for y in the equation z=-\frac{115}{16}+\frac{5}{8}y.

$$z=-\frac{115}{16}+\frac{5}{8}\left(\frac{9}{8}z+\frac{85}{8}\right)$$

Solve z=-\frac{115}{16}+\frac{5}{8}\left(\frac{9}{8}z+\frac{85}{8}\right) for z.

$$z=-\frac{35}{19}$$

Substitute -\frac{35}{19} for z in the equation y=\frac{9}{8}z+\frac{85}{8}.

$$y=\frac{9}{8}\left(-\frac{35}{19}\right)+\frac{85}{8}$$

Calculate y from y=\frac{9}{8}\left(-\frac{35}{19}\right)+\frac{85}{8}.

$$y=\frac{325}{38}$$

Substitute \frac{325}{38} for y and -\frac{35}{19} for z in the equation x=5y-12z-66.

$$x=5\times \frac{325}{38}-12\left(-\frac{35}{19}\right)-66$$

Calculate x from x=5\times \frac{325}{38}-12\left(-\frac{35}{19}\right)-66.

$$x=-\frac{43}{38}$$

The system is now solved.

$$x=-\frac{43}{38} y=\frac{325}{38} z=-\frac{35}{19}$$