Solve {l}{6x+7y+10z=100}{2x+10y+7z=50}{3x-6y-7z=40}

$$\left. \begin{array} { l } { 6 x + 7 y + 10 z = 100 } \\ { 2 x + 10 y + 7 z = 50 } \\ { 3 x - 6 y - 7 z = 40 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{110}{7} = 15\frac{5}{7} \approx 15.714285714$$

$$y = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

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

Short Steps Using Substitution

Solve 6x+7y+10z=100 for x.

$$x=-\frac{7}{6}y-\frac{5}{3}z+\frac{50}{3}$$

Substitute -\frac{7}{6}y-\frac{5}{3}z+\frac{50}{3} for x in the second and third equation.

$$2\left(-\frac{7}{6}y-\frac{5}{3}z+\frac{50}{3}\right)+10y+7z=50 3\left(-\frac{7}{6}y-\frac{5}{3}z+\frac{50}{3}\right)-6y-7z=40$$

Solve these equations for y and z respectively.

$$y=\frac{50}{23}-\frac{11}{23}z z=\frac{5}{6}-\frac{19}{24}y$$

Substitute \frac{50}{23}-\frac{11}{23}z for y in the equation z=\frac{5}{6}-\frac{19}{24}y.

$$z=\frac{5}{6}-\frac{19}{24}\left(\frac{50}{23}-\frac{11}{23}z\right)$$

Solve z=\frac{5}{6}-\frac{19}{24}\left(\frac{50}{23}-\frac{11}{23}z\right) for z.

$$z=-\frac{10}{7}$$

Substitute -\frac{10}{7} for z in the equation y=\frac{50}{23}-\frac{11}{23}z.

$$y=\frac{50}{23}-\frac{11}{23}\left(-\frac{10}{7}\right)$$

Calculate y from y=\frac{50}{23}-\frac{11}{23}\left(-\frac{10}{7}\right).

$$y=\frac{20}{7}$$

Substitute \frac{20}{7} for y and -\frac{10}{7} for z in the equation x=-\frac{7}{6}y-\frac{5}{3}z+\frac{50}{3}.

$$x=-\frac{7}{6}\times \frac{20}{7}-\frac{5}{3}\left(-\frac{10}{7}\right)+\frac{50}{3}$$

Calculate x from x=-\frac{7}{6}\times \frac{20}{7}-\frac{5}{3}\left(-\frac{10}{7}\right)+\frac{50}{3}.

$$x=\frac{110}{7}$$

The system is now solved.

$$x=\frac{110}{7} y=\frac{20}{7} z=-\frac{10}{7}$$