1. **State the problem:** We have an arithmetic sequence with terms $u_3 = 12$ and $u_{10} = 40$. The common difference is $d$. We need to write two equations involving $u_1$ (the first term) and $d$ that represent this information. 2. **Recall the formula for the $n$-th term of an arithmetic sequence:** $$u_n = u_1 + (n-1)d$$ This means each term is the first term plus the common difference multiplied by one less than the term number. 3. **Write the equation for $u_3$:** $$u_3 = u_1 + (3-1)d = u_1 + 2d$$ Given $u_3 = 12$, we have: $$u_1 + 2d = 12$$ 4. **Write the equation for $u_{10}$:** $$u_{10} = u_1 + (10-1)d = u_1 + 9d$$ Given $u_{10} = 40$, we have: $$u_1 + 9d = 40$$ **Final answer:** The two equations are: $$u_1 + 2d = 12$$ $$u_1 + 9d = 40$$