Solve {l}{x+y+z=78}{x=2y}{x+4=3z}

$$\left. \begin{array} { l } { x + y + z = 78 } \\ { x = 2 y } \\ { x + 4 = 3 z } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{460}{11} = 41\frac{9}{11} \approx 41.818181818$$

$$y = \frac{230}{11} = 20\frac{10}{11} \approx 20.909090909$$

$$z = \frac{168}{11} = 15\frac{3}{11} \approx 15.272727273$$

Short Steps Using Substitution

Reorder the equations.

$$x=2y x+y+z=78 x+4=3z$$

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

$$2y+y+z=78 2y+4=3z$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{3}z+26 z=\frac{4}{3}+\frac{2}{3}y$$

Substitute -\frac{1}{3}z+26 for y in the equation z=\frac{4}{3}+\frac{2}{3}y.

$$z=\frac{4}{3}+\frac{2}{3}\left(-\frac{1}{3}z+26\right)$$

Solve z=\frac{4}{3}+\frac{2}{3}\left(-\frac{1}{3}z+26\right) for z.

$$z=\frac{168}{11}$$

Substitute \frac{168}{11} for z in the equation y=-\frac{1}{3}z+26.

$$y=-\frac{1}{3}\times \frac{168}{11}+26$$

Calculate y from y=-\frac{1}{3}\times \frac{168}{11}+26.

$$y=\frac{230}{11}$$

Substitute \frac{230}{11} for y in the equation x=2y.

$$x=2\times \frac{230}{11}$$

Calculate x from x=2\times \frac{230}{11}.

$$x=\frac{460}{11}$$

The system is now solved.

$$x=\frac{460}{11} y=\frac{230}{11} z=\frac{168}{11}$$