Solve {l}{x+y+z=201}{y=1/3x+60}{z=1/3y+30}

$$\left\{ \begin{array} { l } { x + y + z = 201 } \\ { y = \frac { 1 } { 3 } x + 60 } \\ { z = \frac { 1 } { 3 } y + 30 } \end{array} \right.$$

Solve for x, y, z

$$x=63$$

$$y=81$$

$$z=57$$

Short Steps Using Substitution

Reorder the equations.

$$y=\frac{1}{3}x+60 x+y+z=201 z=\frac{1}{3}y+30$$

Substitute \frac{1}{3}x+60 for y in the second and third equation.

$$x+\frac{1}{3}x+60+z=201 z=\frac{1}{3}\left(\frac{1}{3}x+60\right)+30$$

Solve these equations for x and z respectively.

$$x=\frac{423}{4}-\frac{3}{4}z z=50+\frac{1}{9}x$$

Substitute \frac{423}{4}-\frac{3}{4}z for x in the equation z=50+\frac{1}{9}x.

$$z=50+\frac{1}{9}\left(\frac{423}{4}-\frac{3}{4}z\right)$$

Solve z=50+\frac{1}{9}\left(\frac{423}{4}-\frac{3}{4}z\right) for z.

$$z=57$$

Substitute 57 for z in the equation x=\frac{423}{4}-\frac{3}{4}z.

$$x=\frac{423}{4}-\frac{3}{4}\times 57$$

Calculate x from x=\frac{423}{4}-\frac{3}{4}\times 57.

$$x=63$$

Substitute 63 for x in the equation y=\frac{1}{3}x+60.

$$y=\frac{1}{3}\times 63+60$$

Calculate y from y=\frac{1}{3}\times 63+60.

$$y=81$$

The system is now solved.

$$x=63 y=81 z=57$$