Solve {l}{x+y+z=21}{0x+100y+200z=1800}{5x-z=0}

$$\left. \begin{array} { l } { x + y + z = 21 } \\ { 0 x + 100 y + 200 z = 1800 } \\ { 5 x - z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{3}{4}=-0.75$$

$$y = \frac{51}{2} = 25\frac{1}{2} = 25.5$$

$$z = -\frac{15}{4} = -3\frac{3}{4} = -3.75$$

Short Steps Using Substitution

Solve x+y+z=21 for x.

$$x=-y-z+21$$

Substitute -y-z+21 for x in the second and third equation.

$$0\left(-y-z+21\right)+100y+200z=1800 5\left(-y-z+21\right)-z=0$$

Solve these equations for y and z respectively.

$$y=18-2z z=\frac{35}{2}-\frac{5}{6}y$$

Substitute 18-2z for y in the equation z=\frac{35}{2}-\frac{5}{6}y.

$$z=\frac{35}{2}-\frac{5}{6}\left(18-2z\right)$$

Solve z=\frac{35}{2}-\frac{5}{6}\left(18-2z\right) for z.

$$z=-\frac{15}{4}$$

Substitute -\frac{15}{4} for z in the equation y=18-2z.

$$y=18-2\left(-\frac{15}{4}\right)$$

Calculate y from y=18-2\left(-\frac{15}{4}\right).

$$y=\frac{51}{2}$$

Substitute \frac{51}{2} for y and -\frac{15}{4} for z in the equation x=-y-z+21.

$$x=-\frac{51}{2}-\left(-\frac{15}{4}\right)+21$$

Calculate x from x=-\frac{51}{2}-\left(-\frac{15}{4}\right)+21.

$$x=-\frac{3}{4}$$

The system is now solved.

$$x=-\frac{3}{4} y=\frac{51}{2} z=-\frac{15}{4}$$