An investor has invested in three stocks: Stock $1,$ Stock $1,$ and Stock L$.$ Let x$,$ y$,$ and z represent the percentages of the total investment in each stock, respectively. The investor has the following constraints: The total investment adds up to $100\%.$ Stock X has an expected return of $8\%$ per annum, Stock Y has an expected return of $10\%$ per annum, and Stock Z has an expected return of $12\%$ per annum. The investor wants to allocate twice as much in Stock Z as in Stock X$.$ The investment in Stock Y should be $30\%$ of the total investment.$($a$)$ Construct the system of linear equations representing the above constraints.$($b$)$ Determine, using matrix algebra, the percentages of the total investment allocated to each stock that maximize the expected return.