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

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

Solve for x, y, z

$$x=\frac{31}{54}\approx 0.574074074$$

$$y=-\frac{23}{36}\approx -0.638888889$$

$$z=\frac{97}{108}\approx 0.898148148$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

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

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

Solve these equations for y and z respectively.

$$y=\frac{9}{7}-\frac{15}{7}z z=-\frac{5}{3}y-\frac{1}{6}$$

Substitute \frac{9}{7}-\frac{15}{7}z for y in the equation z=-\frac{5}{3}y-\frac{1}{6}.

$$z=-\frac{5}{3}\left(\frac{9}{7}-\frac{15}{7}z\right)-\frac{1}{6}$$

Solve z=-\frac{5}{3}\left(\frac{9}{7}-\frac{15}{7}z\right)-\frac{1}{6} for z.

$$z=\frac{97}{108}$$

Substitute \frac{97}{108} for z in the equation y=\frac{9}{7}-\frac{15}{7}z.

$$y=\frac{9}{7}-\frac{15}{7}\times \frac{97}{108}$$

Calculate y from y=\frac{9}{7}-\frac{15}{7}\times \frac{97}{108}.

$$y=-\frac{23}{36}$$

Substitute -\frac{23}{36} for y and \frac{97}{108} for z in the equation x=3y+5z-2.

$$x=3\left(-\frac{23}{36}\right)+5\times \frac{97}{108}-2$$

Calculate x from x=3\left(-\frac{23}{36}\right)+5\times \frac{97}{108}-2.

$$x=\frac{31}{54}$$

The system is now solved.

$$x=\frac{31}{54} y=-\frac{23}{36} z=\frac{97}{108}$$