Solve {l}{x+y+z=7}{4x-2y-z=3}{2x-y+3z=-1}

$$\left. \begin{array} { l } { x + y + z = 7 } \\ { 4 x - 2 y - z = 3 } \\ { 2 x - y + 3 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{62}{21} = 2\frac{20}{21} \approx 2.952380952$$

$$y = \frac{100}{21} = 4\frac{16}{21} \approx 4.761904762$$

$$z=-\frac{5}{7}\approx -0.714285714$$

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.

$$4\left(-y-z+7\right)-2y-z=3 2\left(-y-z+7\right)-y+3z=-1$$

Solve these equations for y and z respectively.

$$y=\frac{25}{6}-\frac{5}{6}z z=-15+3y$$

Substitute \frac{25}{6}-\frac{5}{6}z for y in the equation z=-15+3y.

$$z=-15+3\left(\frac{25}{6}-\frac{5}{6}z\right)$$

Solve z=-15+3\left(\frac{25}{6}-\frac{5}{6}z\right) for z.

$$z=-\frac{5}{7}$$

Substitute -\frac{5}{7} for z in the equation y=\frac{25}{6}-\frac{5}{6}z.

$$y=\frac{25}{6}-\frac{5}{6}\left(-\frac{5}{7}\right)$$

Calculate y from y=\frac{25}{6}-\frac{5}{6}\left(-\frac{5}{7}\right).

$$y=\frac{100}{21}$$

Substitute \frac{100}{21} for y and -\frac{5}{7} for z in the equation x=-y-z+7.

$$x=-\frac{100}{21}-\left(-\frac{5}{7}\right)+7$$

Calculate x from x=-\frac{100}{21}-\left(-\frac{5}{7}\right)+7.

$$x=\frac{62}{21}$$

The system is now solved.

$$x=\frac{62}{21} y=\frac{100}{21} z=-\frac{5}{7}$$