Solve {l}{x+y+z=12}{x+2y+5z=-23}{x=4y}

$$\left\{ \begin{array} { l } { x + y + z = 12 } \\ { x + 2 y + 5 z = - 23 } \\ { x = 4 y } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{332}{19} = 17\frac{9}{19} \approx 17.473684211$$

$$y = \frac{83}{19} = 4\frac{7}{19} \approx 4.368421053$$

$$z = -\frac{187}{19} = -9\frac{16}{19} \approx -9.842105263$$

Short Steps Using Substitution

Reorder the equations.

$$x=4y x+2y+5z=-23 x+y+z=12$$

Substitute 4y for x in the second and third equation.

$$4y+2y+5z=-23 4y+y+z=12$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{6}z-\frac{23}{6} z=-5y+12$$

Substitute -\frac{5}{6}z-\frac{23}{6} for y in the equation z=-5y+12.

$$z=-5\left(-\frac{5}{6}z-\frac{23}{6}\right)+12$$

Solve z=-5\left(-\frac{5}{6}z-\frac{23}{6}\right)+12 for z.

$$z=-\frac{187}{19}$$

Substitute -\frac{187}{19} for z in the equation y=-\frac{5}{6}z-\frac{23}{6}.

$$y=-\frac{5}{6}\left(-\frac{187}{19}\right)-\frac{23}{6}$$

Calculate y from y=-\frac{5}{6}\left(-\frac{187}{19}\right)-\frac{23}{6}.

$$y=\frac{83}{19}$$

Substitute \frac{83}{19} for y in the equation x=4y.

$$x=4\times \frac{83}{19}$$

Calculate x from x=4\times \frac{83}{19}.

$$x=\frac{332}{19}$$

The system is now solved.

$$x=\frac{332}{19} y=\frac{83}{19} z=-\frac{187}{19}$$