Solve {l}{4x+4y-5z=10}{x-10y+4z=2}{x+15y-2z=0}

$$\left. \begin{array} { l } { 4 x + 4 y - 5 z = 10 } \\ { x - 10 y + 4 z = 2 } \\ { x + 15 y - 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{178}{87} = 2\frac{4}{87} \approx 2.045977011$$

$$y=-\frac{6}{29}\approx -0.206896552$$

$$z=-\frac{46}{87}\approx -0.528735632$$

Short Steps Using Substitution

Reorder the equations.

$$x-10y+4z=2 4x+4y-5z=10 x+15y-2z=0$$

Solve x-10y+4z=2 for x.

$$x=2+10y-4z$$

Substitute 2+10y-4z for x in the second and third equation.

$$4\left(2+10y-4z\right)+4y-5z=10 2+10y-4z+15y-2z=0$$

Solve these equations for y and z respectively.

$$y=\frac{1}{22}+\frac{21}{44}z z=\frac{1}{3}+\frac{25}{6}y$$

Substitute \frac{1}{22}+\frac{21}{44}z for y in the equation z=\frac{1}{3}+\frac{25}{6}y.

$$z=\frac{1}{3}+\frac{25}{6}\left(\frac{1}{22}+\frac{21}{44}z\right)$$

Solve z=\frac{1}{3}+\frac{25}{6}\left(\frac{1}{22}+\frac{21}{44}z\right) for z.

$$z=-\frac{46}{87}$$

Substitute -\frac{46}{87} for z in the equation y=\frac{1}{22}+\frac{21}{44}z.

$$y=\frac{1}{22}+\frac{21}{44}\left(-\frac{46}{87}\right)$$

Calculate y from y=\frac{1}{22}+\frac{21}{44}\left(-\frac{46}{87}\right).

$$y=-\frac{6}{29}$$

Substitute -\frac{6}{29} for y and -\frac{46}{87} for z in the equation x=2+10y-4z.

$$x=2+10\left(-\frac{6}{29}\right)-4\left(-\frac{46}{87}\right)$$

Calculate x from x=2+10\left(-\frac{6}{29}\right)-4\left(-\frac{46}{87}\right).

$$x=\frac{178}{87}$$

The system is now solved.

$$x=\frac{178}{87} y=-\frac{6}{29} z=-\frac{46}{87}$$