Solve {l}{x+y+z=7}{4x-2y-z=8}{2x-y+3z=12}

$$\left. \begin{array} { l } { x + y + z = 7 } \\ { 4 x - 2 y - z = 8 } \\ { 2 x - y + 3 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{23}{7} = 3\frac{2}{7} \approx 3.285714286$$

$$y = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

$$z = \frac{16}{7} = 2\frac{2}{7} \approx 2.285714286$$

Short Steps Using Substitution

Solve x+y+z=7 for x.

$$x=-y-z+7$$

Substitute -y-z+7 for x in the second and third equation.

$$4\left(-y-z+7\right)-2y-z=8 2\left(-y-z+7\right)-y+3z=12$$

Solve these equations for y and z respectively.

$$y=\frac{10}{3}-\frac{5}{6}z z=3y-2$$

Substitute \frac{10}{3}-\frac{5}{6}z for y in the equation z=3y-2.

$$z=3\left(\frac{10}{3}-\frac{5}{6}z\right)-2$$

Solve z=3\left(\frac{10}{3}-\frac{5}{6}z\right)-2 for z.

$$z=\frac{16}{7}$$

Substitute \frac{16}{7} for z in the equation y=\frac{10}{3}-\frac{5}{6}z.

$$y=\frac{10}{3}-\frac{5}{6}\times \frac{16}{7}$$

Calculate y from y=\frac{10}{3}-\frac{5}{6}\times \frac{16}{7}.

$$y=\frac{10}{7}$$

Substitute \frac{10}{7} for y and \frac{16}{7} for z in the equation x=-y-z+7.

$$x=-\frac{10}{7}-\frac{16}{7}+7$$

Calculate x from x=-\frac{10}{7}-\frac{16}{7}+7.

$$x=\frac{23}{7}$$

The system is now solved.

$$x=\frac{23}{7} y=\frac{10}{7} z=\frac{16}{7}$$