Solve {l}{+2x-3y+z=-9}{-4x+y+2z=-1}{-5x-y-4z=-6}

$$\left\{ \begin{array} { l } { + 2 x - 3 y + z = - 9 } \\ { - 4 x + y + 2 z = - 1 } \\ { - 5 x - y - 4 z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{73}{83}\approx 0.879518072$$

$$y = \frac{285}{83} = 3\frac{36}{83} \approx 3.43373494$$

$$z=-\frac{38}{83}\approx -0.457831325$$

Short Steps Using Substitution

Solve 2x-3y+z=-9 for z.

$$z=-2x+3y-9$$

Substitute -2x+3y-9 for z in the second and third equation.

$$-4x+y+2\left(-2x+3y-9\right)=-1 -5x-y-4\left(-2x+3y-9\right)=-6$$

Solve these equations for y and x respectively.

$$y=\frac{8}{7}x+\frac{17}{7} x=\frac{13}{3}y-14$$

Substitute \frac{8}{7}x+\frac{17}{7} for y in the equation x=\frac{13}{3}y-14.

$$x=\frac{13}{3}\left(\frac{8}{7}x+\frac{17}{7}\right)-14$$

Solve x=\frac{13}{3}\left(\frac{8}{7}x+\frac{17}{7}\right)-14 for x.

$$x=\frac{73}{83}$$

Substitute \frac{73}{83} for x in the equation y=\frac{8}{7}x+\frac{17}{7}.

$$y=\frac{8}{7}\times \frac{73}{83}+\frac{17}{7}$$

Calculate y from y=\frac{8}{7}\times \frac{73}{83}+\frac{17}{7}.

$$y=\frac{285}{83}$$

Substitute \frac{285}{83} for y and \frac{73}{83} for x in the equation z=-2x+3y-9.

$$z=-2\times \frac{73}{83}+3\times \frac{285}{83}-9$$

Calculate z from z=-2\times \frac{73}{83}+3\times \frac{285}{83}-9.

$$z=-\frac{38}{83}$$

The system is now solved.

$$x=\frac{73}{83} y=\frac{285}{83} z=-\frac{38}{83}$$