Solve {c}{0.27x+0.02y+0.02z=200}{0.17x+0.07y+0.8z=170}{x+y+z=1000}

$$\left. \begin{array} { c } { 0.27 x + 0.02 y + 0.02 z = 200 } \\ { 0.17 x + 0.07 y + 0.8 z = 170 } \\ { x + y + z = 1000 } \end{array} \right.$$

Solve for x, y, z

$$x=720$$

$$y = \frac{17640}{73} = 241\frac{47}{73} \approx 241.643835616$$

$$z = \frac{2800}{73} = 38\frac{26}{73} \approx 38.356164384$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+z=1000 0.17x+0.07y+0.8z=170 0.27x+0.02y+0.02z=200$$

Solve x+y+z=1000 for x.

$$x=-y-z+1000$$

Substitute -y-z+1000 for x in the second and third equation.

$$0.17\left(-y-z+1000\right)+0.07y+0.8z=170 0.27\left(-y-z+1000\right)+0.02y+0.02z=200$$

Solve these equations for y and z respectively.

$$y=6.3z z=-y+280$$

Substitute 6.3z for y in the equation z=-y+280.

$$z=-6.3z+280$$

Solve z=-6.3z+280 for z.

$$z=\frac{2800}{73}$$

Substitute \frac{2800}{73} for z in the equation y=6.3z.

$$y=6.3\times \frac{2800}{73}$$

Calculate y from y=6.3\times \frac{2800}{73}.

$$y=\frac{17640}{73}$$

Substitute \frac{17640}{73} for y and \frac{2800}{73} for z in the equation x=-y-z+1000.

$$x=-\frac{17640}{73}-\frac{2800}{73}+1000$$

Calculate x from x=-\frac{17640}{73}-\frac{2800}{73}+1000.

$$x=720$$

The system is now solved.

$$x=720 y=\frac{17640}{73} z=\frac{2800}{73}$$