Solve {l}{3x+2y+z=1}{2x+6y+2z=1}{2x+8y+3z=9}

$$\left. \begin{array} { l } { 3 x + 2 y + z = 1 } \\ { 2 x + 6 y + 2 z = 1 } \\ { 2 x + 8 y + 3 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$y = -\frac{31}{6} = -5\frac{1}{6} \approx -5.166666667$$

$$z = \frac{55}{3} = 18\frac{1}{3} \approx 18.333333333$$

Short Steps Using Substitution

Solve 3x+2y+z=1 for z.

$$z=-3x-2y+1$$

Substitute -3x-2y+1 for z in the second and third equation.

$$2x+6y+2\left(-3x-2y+1\right)=1 2x+8y+3\left(-3x-2y+1\right)=9$$

Solve these equations for y and x respectively.

$$y=2x-\frac{1}{2} x=-\frac{6}{7}+\frac{2}{7}y$$

Substitute 2x-\frac{1}{2} for y in the equation x=-\frac{6}{7}+\frac{2}{7}y.

$$x=-\frac{6}{7}+\frac{2}{7}\left(2x-\frac{1}{2}\right)$$

Solve x=-\frac{6}{7}+\frac{2}{7}\left(2x-\frac{1}{2}\right) for x.

$$x=-\frac{7}{3}$$

Substitute -\frac{7}{3} for x in the equation y=2x-\frac{1}{2}.

$$y=2\left(-\frac{7}{3}\right)-\frac{1}{2}$$

Calculate y from y=2\left(-\frac{7}{3}\right)-\frac{1}{2}.

$$y=-\frac{31}{6}$$

Substitute -\frac{31}{6} for y and -\frac{7}{3} for x in the equation z=-3x-2y+1.

$$z=-3\left(-\frac{7}{3}\right)-2\left(-\frac{31}{6}\right)+1$$

Calculate z from z=-3\left(-\frac{7}{3}\right)-2\left(-\frac{31}{6}\right)+1.

$$z=\frac{55}{3}$$

The system is now solved.

$$x=-\frac{7}{3} y=-\frac{31}{6} z=\frac{55}{3}$$