Solve {l}{3x+4y+8z=1}{2z+3y+x=2}{z+2z+2y=3}

$$\left. \begin{array} { l } { 3 x + 4 y + 8 z = 1 } \\ { 2 z + 3 y + x = 2 } \\ { z + 2 z + 2 y = 3 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{35}{19} = -1\frac{16}{19} \approx -1.842105263$$

$$y = \frac{21}{19} = 1\frac{2}{19} \approx 1.105263158$$

$$z=\frac{5}{19}\approx 0.263157895$$

Short Steps Using Substitution

Reorder the equations.

$$2z+3y+x=2 3x+4y+8z=1 z+2z+2y=3$$

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

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

Substitute -2z-3y+2 for x in the equation 3x+4y+8z=1.

$$3\left(-2z-3y+2\right)+4y+8z=1$$

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

$$y=\frac{2}{5}z+1 z=-\frac{2}{3}y+1$$

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

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

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

$$z=\frac{5}{19}$$

Substitute \frac{5}{19} for z in the equation y=\frac{2}{5}z+1.

$$y=\frac{2}{5}\times \frac{5}{19}+1$$

Calculate y from y=\frac{2}{5}\times \frac{5}{19}+1.

$$y=\frac{21}{19}$$

Substitute \frac{21}{19} for y and \frac{5}{19} for z in the equation x=-2z-3y+2.

$$x=-2\times \frac{5}{19}-3\times \frac{21}{19}+2$$

Calculate x from x=-2\times \frac{5}{19}-3\times \frac{21}{19}+2.

$$x=-\frac{35}{19}$$

The system is now solved.

$$x=-\frac{35}{19} y=\frac{21}{19} z=\frac{5}{19}$$