Let x be the number of associates designated.
Let y be the number of partners designated.
If there are a total of 13 lawyers assigned to the case, then this represents the sum of associates and partners:
![x+y=13...(1)](https://img.qammunity.org/2023/formulas/mathematics/college/y8cnyh9j4j6r9xoqtibovjl6zhqkrwxs00.png)
Additionally, we know that the daily rate charged to the client for each associate is $400 and the daily rate for each partner is $1000. Then, the daily rate for all associates is 400x and the daily rate for all partners is 1000y. Knowing that the total daily charge to the client is $8800, then:
![400x+1000y=8800](https://img.qammunity.org/2023/formulas/mathematics/college/13vfoza694iqfchxrf5n52fqa9ncpfrjr6.png)
This equation can be simplified. Dividing by 200 on both sides:
![2x+5y=44...(2)](https://img.qammunity.org/2023/formulas/mathematics/college/lxywnnhhd403xw6zjghdjo7nct5ttsuzz3.png)
From (1) and (2), the system of equations is:
![\begin{cases}x+y={13} \\ 2x+5y={44}\end{cases}](https://img.qammunity.org/2023/formulas/mathematics/college/5906mj056kp6kv2fiva5ql4in6q0qzi89z.png)