Solve {l}{3x+2y+z=4}{x+y+3z=6}{2x-y+6z=10}

$$\left. \begin{array}{l}{ 3 x + 2 y + z = 4 }\\{ x + y + 3 z = 6 }\\{ 2 x - y + 6 z = 10 }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=\frac{2}{3}\approx 0.666666667$$

$$z = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Short Steps Using Substitution

Solve 3x+2y+z=4 for z.

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

Substitute -3x-2y+4 for z in the second and third equation.

$$x+y+3\left(-3x-2y+4\right)=6 2x-y+6\left(-3x-2y+4\right)=10$$

Solve these equations for y and x respectively.

$$y=\frac{6}{5}-\frac{8}{5}x x=-\frac{13}{16}y+\frac{7}{8}$$

Substitute \frac{6}{5}-\frac{8}{5}x for y in the equation x=-\frac{13}{16}y+\frac{7}{8}.

$$x=-\frac{13}{16}\left(\frac{6}{5}-\frac{8}{5}x\right)+\frac{7}{8}$$

Solve x=-\frac{13}{16}\left(\frac{6}{5}-\frac{8}{5}x\right)+\frac{7}{8} for x.

$$x=\frac{1}{3}$$

Substitute \frac{1}{3} for x in the equation y=\frac{6}{5}-\frac{8}{5}x.

$$y=\frac{6}{5}-\frac{8}{5}\times \frac{1}{3}$$

Calculate y from y=\frac{6}{5}-\frac{8}{5}\times \frac{1}{3}.

$$y=\frac{2}{3}$$

Substitute \frac{2}{3} for y and \frac{1}{3} for x in the equation z=-3x-2y+4.

$$z=-3\times \frac{1}{3}-2\times \frac{2}{3}+4$$

Calculate z from z=-3\times \frac{1}{3}-2\times \frac{2}{3}+4.

$$z=\frac{5}{3}$$

The system is now solved.

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