1. **Problem statement:** Among 100 students, (75 - x) study physics, (50 - x) study accounts, and 25 study neither. No student studies both subjects. Find the value of $x$, the number studying physics, and the number studying accounts. 2. **Understanding the problem:** Since 25 students study neither subject, the remaining students studying either physics or accounts are $100 - 25 = 75$. 3. **Key rule:** No student studies both subjects, so the sets are disjoint. Therefore, the total number studying physics plus the total number studying accounts equals 75. 4. **Set up the equation:** $$ (75 - x) + (50 - x) = 75 $$ 5. **Simplify:** $$ 75 - x + 50 - x = 75 $$ $$ 125 - 2x = 75 $$ 6. **Solve for $x$:** $$ 125 - 2x = 75 $$ $$ -2x = 75 - 125 $$ $$ -2x = -50 $$ $$ x = \frac{-50}{-2} = 25 $$ 7. **Find the number studying physics:** $$ 75 - x = 75 - 25 = 50 $$ 8. **Find the number studying accounts:** $$ 50 - x = 50 - 25 = 25 $$ **Final answers:** - $x = 25$ - Number studying physics = 50 - Number studying accounts = 25