Solve {l}{-2x+y+6z=18}{5x+8z=-16}{3x+2y-10z=-3}

$$\left. \begin{array} { l } { - 2 x + y + 6 z = 18 } \\ { 5 x + 8 z = - 16 } \\ { 3 x + 2 y - 10 z = - 3 } \end{array} \right.$$

Solve for x, y, z

$$x=-4$$

$$y=7$$

$$z=\frac{1}{2}=0.5$$

Short Steps Using Substitution

Solve -2x+y+6z=18 for y.

$$y=18+2x-6z$$

Substitute 18+2x-6z for y in the equation 3x+2y-10z=-3.

$$3x+2\left(18+2x-6z\right)-10z=-3$$

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

$$x=-\frac{16}{5}-\frac{8}{5}z z=\frac{7}{22}x+\frac{39}{22}$$

Substitute -\frac{16}{5}-\frac{8}{5}z for x in the equation z=\frac{7}{22}x+\frac{39}{22}.

$$z=\frac{7}{22}\left(-\frac{16}{5}-\frac{8}{5}z\right)+\frac{39}{22}$$

Solve z=\frac{7}{22}\left(-\frac{16}{5}-\frac{8}{5}z\right)+\frac{39}{22} for z.

$$z=\frac{1}{2}$$

Substitute \frac{1}{2} for z in the equation x=-\frac{16}{5}-\frac{8}{5}z.

$$x=-\frac{16}{5}-\frac{8}{5}\times \frac{1}{2}$$

Calculate x from x=-\frac{16}{5}-\frac{8}{5}\times \frac{1}{2}.

$$x=-4$$

Substitute -4 for x and \frac{1}{2} for z in the equation y=18+2x-6z.

$$y=18+2\left(-4\right)-6\times \frac{1}{2}$$

Calculate y from y=18+2\left(-4\right)-6\times \frac{1}{2}.

$$y=7$$

The system is now solved.

$$x=-4 y=7 z=\frac{1}{2}$$