In a decentralized finance $($DeFi$)$ ecosystem, three types of cryptocurrencies, Crypto A$,$ Crypto $B,$ and Crypto $C,$ are being traded on a platform. Investors are interested in diversifying their portfolios between these three cryptocurrencies to maximize their potential returns. The current exchange rates and investment constraints are as follows:

$1$ unit of Crypto $A$ is equivalent to $$50.$

$1$ unit of Crypto $B$ is equivalent to $$100.$

$1$ unit of Crypto $C$ is equivalent to $$75.$

The investor wants to invest $$10,000$ in these cryptocurrencies and maintain $150$ cryptos in total.

$($a$)$ Write down and solve the system of equations representing the investment scenario.

$($b$)$ Find out how much money the investor should invest in each cryptocurrency to maximize their investment in Crypto C$.$