Solve {l}{3x+5y-4z=0}{-6x+10y-4z=-9}{15y+8z=-4}

$$\left\{ \begin{array}{l}{ 3 x + 5 y - 4 z = 0 }\\{ - 6 x + 10 y - 4 z = - 9 }\\{ 15 y + 8 z = - 4 }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{41}{51}\approx 0.803921569$$

$$y=-\frac{6}{17}\approx -0.352941176$$

$$z=\frac{11}{68}\approx 0.161764706$$

Short Steps Using Substitution

Solve 3x+5y-4z=0 for x.

$$x=-\frac{5}{3}y+\frac{4}{3}z$$

Substitute -\frac{5}{3}y+\frac{4}{3}z for x in the equation -6x+10y-4z=-9.

$$-6\left(-\frac{5}{3}y+\frac{4}{3}z\right)+10y-4z=-9$$

Solve the second equation for y and the third equation for z.

$$y=-\frac{9}{20}+\frac{3}{5}z z=-\frac{15}{8}y-\frac{1}{2}$$

Substitute -\frac{9}{20}+\frac{3}{5}z for y in the equation z=-\frac{15}{8}y-\frac{1}{2}.

$$z=-\frac{15}{8}\left(-\frac{9}{20}+\frac{3}{5}z\right)-\frac{1}{2}$$

Solve z=-\frac{15}{8}\left(-\frac{9}{20}+\frac{3}{5}z\right)-\frac{1}{2} for z.

$$z=\frac{11}{68}$$

Substitute \frac{11}{68} for z in the equation y=-\frac{9}{20}+\frac{3}{5}z.

$$y=-\frac{9}{20}+\frac{3}{5}\times \frac{11}{68}$$

Calculate y from y=-\frac{9}{20}+\frac{3}{5}\times \frac{11}{68}.

$$y=-\frac{6}{17}$$

Substitute -\frac{6}{17} for y and \frac{11}{68} for z in the equation x=-\frac{5}{3}y+\frac{4}{3}z.

$$x=-\frac{5}{3}\left(-\frac{6}{17}\right)+\frac{4}{3}\times \frac{11}{68}$$

Calculate x from x=-\frac{5}{3}\left(-\frac{6}{17}\right)+\frac{4}{3}\times \frac{11}{68}.

$$x=\frac{41}{51}$$

The system is now solved.

$$x=\frac{41}{51} y=-\frac{6}{17} z=\frac{11}{68}$$