Solve {l}{3x+7y+z=0}{x+5y+z=10}{x+6y-2z=7}

$$\left\{ \begin{array} { l } { 3 x + 7 y + z = 0 } \\ { x + 5 y + z = 10 } \\ { x + 6 y - 2 z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{107}{13} = -8\frac{3}{13} \approx -8.230769231$$

$$y = \frac{42}{13} = 3\frac{3}{13} \approx 3.230769231$$

$$z = \frac{27}{13} = 2\frac{1}{13} \approx 2.076923077$$

Short Steps Using Substitution

Solve 3x+7y+z=0 for z.

$$z=-3x-7y$$

Substitute -3x-7y for z in the second and third equation.

$$x+5y-3x-7y=10 x+6y-2\left(-3x-7y\right)=7$$

Solve these equations for y and x respectively.

$$y=-5-x x=-\frac{20}{7}y+1$$

Substitute -5-x for y in the equation x=-\frac{20}{7}y+1.

$$x=-\frac{20}{7}\left(-5-x\right)+1$$

Solve x=-\frac{20}{7}\left(-5-x\right)+1 for x.

$$x=-\frac{107}{13}$$

Substitute -\frac{107}{13} for x in the equation y=-5-x.

$$y=-5-\left(-\frac{107}{13}\right)$$

Calculate y from y=-5-\left(-\frac{107}{13}\right).

$$y=\frac{42}{13}$$

Substitute \frac{42}{13} for y and -\frac{107}{13} for x in the equation z=-3x-7y.

$$z=-3\left(-\frac{107}{13}\right)-7\times \frac{42}{13}$$

Calculate z from z=-3\left(-\frac{107}{13}\right)-7\times \frac{42}{13}.

$$z=\frac{27}{13}$$

The system is now solved.

$$x=-\frac{107}{13} y=\frac{42}{13} z=\frac{27}{13}$$