Solve {l}{y=2x+6+z}{z=3y+7x}{x=y+38}

$$\left. \begin{array} { l } { y = 2 x + 6 + z } \\ { z = 3 y + 7 x } \\ { x = y + 38 } \end{array} \right.$$

Solve for y, x, z

$$x = \frac{70}{11} = 6\frac{4}{11} \approx 6.363636364$$

$$y = -\frac{348}{11} = -31\frac{7}{11} \approx -31.636363636$$

$$z = -\frac{554}{11} = -50\frac{4}{11} \approx -50.363636364$$

Short Steps Using Substitution

Substitute 2x+6+z for y in the second and third equation.

$$z=3\left(2x+6+z\right)+7x x=2x+6+z+38$$

Solve these equations for x and z respectively.

$$x=-\frac{18}{13}-\frac{2}{13}z z=-44-x$$

Substitute -\frac{18}{13}-\frac{2}{13}z for x in the equation z=-44-x.

$$z=-44-\left(-\frac{18}{13}-\frac{2}{13}z\right)$$

Solve z=-44-\left(-\frac{18}{13}-\frac{2}{13}z\right) for z.

$$z=-\frac{554}{11}$$

Substitute -\frac{554}{11} for z in the equation x=-\frac{18}{13}-\frac{2}{13}z.

$$x=-\frac{18}{13}-\frac{2}{13}\left(-\frac{554}{11}\right)$$

Calculate x from x=-\frac{18}{13}-\frac{2}{13}\left(-\frac{554}{11}\right).

$$x=\frac{70}{11}$$

Substitute \frac{70}{11} for x and -\frac{554}{11} for z in the equation y=2x+6+z.

$$y=2\times \frac{70}{11}+6-\frac{554}{11}$$

Calculate y from y=2\times \frac{70}{11}+6-\frac{554}{11}.

$$y=-\frac{348}{11}$$

The system is now solved.

$$y=-\frac{348}{11} x=\frac{70}{11} z=-\frac{554}{11}$$