Solve {l}{x+y+z=20}{x=2y+1}{y/3=z/2}

$$\left\{ \begin{array} { l } { x + y + z = 20 } \\ { x = 2 y + 1 } \\ { \frac { y } { 3 } = \frac { z } { 2 } } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{125}{11} = 11\frac{4}{11} \approx 11.363636364$$

$$y = \frac{57}{11} = 5\frac{2}{11} \approx 5.181818182$$

$$z = \frac{38}{11} = 3\frac{5}{11} \approx 3.454545455$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$x+y+z=20 x=2y+1 2y=3z$$

Reorder the equations.

$$x=2y+1 x+y+z=20 2y=3z$$

Substitute 2y+1 for x in the equation x+y+z=20.

$$2y+1+y+z=20$$

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

$$y=\frac{19}{3}-\frac{1}{3}z z=\frac{2}{3}y$$

Substitute \frac{19}{3}-\frac{1}{3}z for y in the equation z=\frac{2}{3}y.

$$z=\frac{2}{3}\left(\frac{19}{3}-\frac{1}{3}z\right)$$

Solve z=\frac{2}{3}\left(\frac{19}{3}-\frac{1}{3}z\right) for z.

$$z=\frac{38}{11}$$

Substitute \frac{38}{11} for z in the equation y=\frac{19}{3}-\frac{1}{3}z.

$$y=\frac{19}{3}-\frac{1}{3}\times \frac{38}{11}$$

Calculate y from y=\frac{19}{3}-\frac{1}{3}\times \frac{38}{11}.

$$y=\frac{57}{11}$$

Substitute \frac{57}{11} for y in the equation x=2y+1.

$$x=2\times \frac{57}{11}+1$$

Calculate x from x=2\times \frac{57}{11}+1.

$$x=\frac{125}{11}$$

The system is now solved.

$$x=\frac{125}{11} y=\frac{57}{11} z=\frac{38}{11}$$