Solve {l}{10-frac{x_{0}}{2000}-2-frac{x_{0}}{8000}={(10-frac{30000-x_{0}}{30000}-2-frac{30000-x_{0}}{8000})}1}{a=1^-1}{text{Solvefor}btext{where}}{b=a}

$$\left. \begin{array} { l } { 10 - \frac{x_{0}}{2000} - 2 - \frac{x_{0}}{8000} = {(10 - \frac{30000 - x_{0}}{30000} - 2 - \frac{30000 - x_{0}}{8000})} 1 }\\ { a = 1 ^ {-1} }\\ { \text{Solve for } b \text{ where} } \\ { b = a } \end{array} \right.$$

Solve for x_0, a, b

$$b=1$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 120000, the least common multiple of 2000,8000,30000.

$$1200000-60x_{0}-240000-15x_{0}=120000\left(10-\frac{30000-x_{0}}{30000}-2-\frac{30000-x_{0}}{8000}\right)\times 1$$

Subtract 240000 from 1200000 to get 960000.

$$960000-60x_{0}-15x_{0}=120000\left(10-\frac{30000-x_{0}}{30000}-2-\frac{30000-x_{0}}{8000}\right)\times 1$$

Combine -60x_{0} and -15x_{0} to get -75x_{0}.

$$960000-75x_{0}=120000\left(10-\frac{30000-x_{0}}{30000}-2-\frac{30000-x_{0}}{8000}\right)\times 1$$

Subtract 2 from 10 to get 8.

$$960000-75x_{0}=120000\left(8-\frac{30000-x_{0}}{30000}-\frac{30000-x_{0}}{8000}\right)\times 1$$

Combine -\frac{30000-x_{0}}{30000} and -\frac{30000-x_{0}}{8000} to get -\frac{19}{120000}\left(30000-x_{0}\right).

$$960000-75x_{0}=120000\left(8-\frac{19}{120000}\left(30000-x_{0}\right)\right)\times 1$$

Multiply 120000 and 1 to get 120000.

$$960000-75x_{0}=120000\left(8-\frac{19}{120000}\left(30000-x_{0}\right)\right)$$

Use the distributive property to multiply -\frac{19}{120000} by 30000-x_{0}.

$$960000-75x_{0}=120000\left(8-\frac{19}{4}+\frac{19}{120000}x_{0}\right)$$

Subtract \frac{19}{4} from 8 to get \frac{13}{4}.

$$960000-75x_{0}=120000\left(\frac{13}{4}+\frac{19}{120000}x_{0}\right)$$

Use the distributive property to multiply 120000 by \frac{13}{4}+\frac{19}{120000}x_{0}.

$$960000-75x_{0}=390000+19x_{0}$$

Subtract 19x_{0} from both sides.

$$960000-75x_{0}-19x_{0}=390000$$

Combine -75x_{0} and -19x_{0} to get -94x_{0}.

$$960000-94x_{0}=390000$$

Subtract 960000 from both sides.

$$-94x_{0}=390000-960000$$

Subtract 960000 from 390000 to get -570000.

$$-94x_{0}=-570000$$

Divide both sides by -94.

$$x_{0}=\frac{-570000}{-94}$$

Reduce the fraction \frac{-570000}{-94} to lowest terms by extracting and canceling out -2.

$$x_{0}=\frac{285000}{47}$$

Consider the second equation. Calculate 1 to the power of -1 and get 1.

$$a=1$$

Consider the third equation. Insert the known values of variables into the equation.

$$b=1$$

The system is now solved.

$$x_{0}=\frac{285000}{47} a=1 b=1$$