The sum of the two digits of a number is 16 The number formed by reversing the digits is 18 more than the original number. Determine the original number. Let \( t= \) the tens digit, \( u= \) the units digit, and \( u+t=16 \). Which of the following equations would complete the system? \( 9 \mathrm{u}-9 \mathrm{t}=18 \) \( 9 \mathrm{t}-9 \mathrm{u}=18 \) \( \mathrm{tu}=\mathrm{ut}+18 \)

To solve this problem, we first define the original number as \( 10t + u \) and the reversed number as \( 10u + t \). The condition states that the reversed number is 18 more than the original, which can be expressed mathematically as: \[ 10u + t = (10t + u) + 18 \] Now, simplifying this equation will lead us to an equation that relates \( t \) and \( u \). After reordering, we ultimately arrive at: \[ 9u - 9t = 18 \] Rearranging it from here gives us the equation presented. The digits with a sum of 16, \( u + t = 16 \), is a perfect counterbalance to the 9's difference equation. Adding these two core insights together will help you find the values of \( t \) and \( u \). To apply this in real-life scenarios, understanding how to set up equations based on relationships between numbers can help in solving puzzles or even in designing algorithms for games and apps that require numeric manipulation. It’s effectively a blend of logic and creativity! If you're interested in diving deeper into concepts like systems of equations, consider exploring algebraic resources or online tutorials that break down problem sets step by step. Websites like Khan Academy or books on elementary algebra can be quite enriching for further learning!