The sum of the two digits of a number is 9 . If the tens digit is one-half the units digit, what is the number? Let \( t= \) the tens digit, \( u= \) the units digit, and \( t+u=9 \). Which of the following equations would complete the system? \( \mathrm{t}=\frac{1}{2} \mathrm{u}=\frac{1}{2} \mathrm{t} \)

Let's break down the problem step by step. 1. **Extract Known Conditions:** - The sum of the two digits of a number is 9: \( t + u = 9 \). - The tens digit \( t \) is one-half the units digit \( u \): \( t = \frac{1}{2} u \). 2. **Set Up the System of Equations:** We have the following two equations: - \( t + u = 9 \) (1) - \( t = \frac{1}{2} u \) (2) 3. **Substitute Equation (2) into Equation (1):** We can substitute \( t \) from equation (2) into equation (1): \[ \frac{1}{2} u + u = 9 \] 4. **Combine Like Terms:** Combine the terms on the left side: \[ \frac{1}{2} u + \frac{2}{2} u = 9 \] \[ \frac{3}{2} u = 9 \] 5. **Solve for \( u \):** Multiply both sides by \( \frac{2}{3} \): \[ u = 9 \cdot \frac{2}{3} = 6 \] 6. **Find \( t \):** Now substitute \( u = 6 \) back into equation (2) to find \( t \): \[ t = \frac{1}{2} \cdot 6 = 3 \] 7. **Conclusion:** The tens digit \( t \) is 3 and the units digit \( u \) is 6. Therefore, the number is: \[ 36 \] The equation that would complete the system is \( t = \frac{1}{2} u \). Thus, the final answer is the number \( 36 \).