Before paying employee bonuses and state and federal taxes, a company earns profits of $60,000. The company pays employees a bonus equal to 5% of after-tax profits. State tax is 5% of profits (after bonuses are paid). Finally, federal tax is 40% of profits (after bonuses and state tax are paid). Determine a linear equation system to find the amounts paid in bonuses, state tax, and federal tax.

[tex]\left[\begin{array}{ccc}1&0.05&0.05\\0.05&1&0\\0.4&0.4&1\end{array}\right] *\left[\begin{array}{c} x_1\\x_2\\x_3\end{array}\right] = \left[\begin{array}{c} 3000\\3000\\24000\end{array}\right][/tex]

Step-by-step explanation:

Given:

- Profits earned = $ 60, 000

- Bonus after tax = 5%

- State tax after bonus is = 5%

- Federal tax after bonus and state tax is = 40%

Find:

A linear equation system to find the amounts paid in bonuses, state tax, and federal tax.

Solution:

Define variables:

- x_1 : amount paid for bonuses

- x_2: amount paid for state tax

- x_3: amount paid for federal tax

- Since bonuses are paid after taxes at 5 %:

x_1 = 0.05(60,000 - x_2 - x_3)

3000 = x_1 + 0.5x_2 + 0.5x_3

- State taxes are paid after bonuses:

x_2 = 0.05*(60000 - x_1)

x_2 + 0.05x_2 = 3000

- Finally federal tax are paid:

x_3 = 0.4*(60,000 - x_1 - x_2)

x_3 + 0.4x_2 + 0.4x_3 = 24,000

- Therefore the system is:

x_1 + 0.5x_2 + 0.5x_3= 3000

x_2 + 0.05x_2 = 3000

x_3 + 0.4x_2 + 0.4x_3 = 24,000

Therefore the system can be expressed as an equation below:

[tex]\left[\begin{array}{ccc}1&0.05&0.05\\0.05&1&0\\0.4&0.4&1\end{array}\right] *\left[\begin{array}{c} x_1\\x_2\\x_3\end{array}\right] = \left[\begin{array}{c} 3000\\3000\\24000\end{array}\right][/tex]