Solve {l}{x+y+z=7}{-14y-7/1z=0}{3x+6y-2z=0}

$$\left\{ \begin{array} { l } { x + y + z = 7 } \\ { - 14 y - \frac { 7 } { 1 } z = 0 } \\ { 3 x + 6 y - 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{70}{13} = 5\frac{5}{13} \approx 5.384615385$$

$$y = -\frac{21}{13} = -1\frac{8}{13} \approx -1.615384615$$

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

Short Steps Using Substitution

Solve x+y+z=7 for x.

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

Substitute -y-z+7 for x in the equation 3x+6y-2z=0.

$$3\left(-y-z+7\right)+6y-2z=0$$

Solve the second equation for y and the third equation for z.

$$y=-\frac{1}{2}z z=\frac{21}{5}+\frac{3}{5}y$$

Substitute -\frac{1}{2}z for y in the equation z=\frac{21}{5}+\frac{3}{5}y.

$$z=\frac{21}{5}+\frac{3}{5}\left(-\frac{1}{2}\right)z$$

Solve z=\frac{21}{5}+\frac{3}{5}\left(-\frac{1}{2}\right)z for z.

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

Substitute \frac{42}{13} for z in the equation y=-\frac{1}{2}z.

$$y=-\frac{1}{2}\times \frac{42}{13}$$

Calculate y from y=-\frac{1}{2}\times \frac{42}{13}.

$$y=-\frac{21}{13}$$

Substitute -\frac{21}{13} for y and \frac{42}{13} for z in the equation x=-y-z+7.

$$x=-\left(-\frac{21}{13}\right)-\frac{42}{13}+7$$

Calculate x from x=-\left(-\frac{21}{13}\right)-\frac{42}{13}+7.

$$x=\frac{70}{13}$$

The system is now solved.

$$x=\frac{70}{13} y=-\frac{21}{13} z=\frac{42}{13}$$