Does A Two-digit Number Exist Such That The Digits Sum To 9, And When The Digits Are Reversed, The Resulting Number Is 9 Greater Than The Original Number? Identify The System Of Equations That Models The Given Scenario.$[ \begin{array}{l} t + U =
Introduction
In the realm of mathematics, there exist numerous problems that seem simple yet lead to profound and intriguing solutions. One such problem is the existence of a two-digit number where the sum of its digits equals 9, and when the digits are reversed, the resulting number is 9 greater than the original number. In this article, we will delve into the world of mathematics to identify the system of equations that models this given scenario.
Understanding the Problem
Let's break down the problem into its core components. We are looking for a two-digit number, which can be represented as 10t + u, where t is the tens digit and u is the units digit. The sum of the digits is given as t + u = 9. When the digits are reversed, the new number becomes 10u + t. According to the problem, this new number is 9 greater than the original number, which can be expressed as 10u + t = (10t + u) + 9.
Formulating the System of Equations
To model this scenario mathematically, we need to formulate a system of equations. Let's start by expressing the given conditions as equations.
- The sum of the digits is 9: t + u = 9
- The new number is 9 greater than the original number: 10u + t = 10t + u + 9
We can simplify the second equation by combining like terms: 10u + t - 10t - u = 9, which further simplifies to -9t + 9u = 9.
Simplifying the System of Equations
To make the system of equations more manageable, let's simplify the second equation by dividing both sides by -9: t - u = -1.
Now we have two equations:
- t + u = 9
- t - u = -1
Solving the System of Equations
We can solve this system of equations using the method of substitution or elimination. Let's use the elimination method. If we add the two equations together, the variable u will be eliminated: (t + u) + (t - u) = 9 + (-1), which simplifies to 2t = 8.
Finding the Value of t
To find the value of t, we can divide both sides of the equation by 2: t = 8/2, which simplifies to t = 4.
Finding the Value of u
Now that we have the value of t, we can substitute it into one of the original equations to find the value of u. Let's use the first equation: t + u = 9. Substituting t = 4, we get 4 + u = 9, which simplifies to u = 5.
Conclusion
In this article, we have identified the system of equations that models the given scenario. We have also solved the system of equations to find the values of t and u. The values of t and u are 4 and 5, respectively. This means that the two-digit number that satisfies the given conditions is 45.
The Final Answer
The final answer is 45.
Discussion
The problem of finding a two-digit number where the sum of its digits equals 9 and when the digits are reversed, the resulting number is 9 greater than the original number is a classic example of a mathematical puzzle. The solution to this problem involves formulating a system of equations, simplifying the system, and solving for the values of t and u. The values of t and u are 4 and 5, respectively, which means that the two-digit number that satisfies the given conditions is 45.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Mathematics for the Nonmathematician" by Morris Kline
Appendix
The following is a list of the steps involved in solving the system of equations:
- Formulate the system of equations
- Simplify the system of equations
- Solve for the values of t and u
- Substitute the values of t and u into one of the original equations to verify the solution
Introduction
In our previous article, we explored the problem of finding a two-digit number where the sum of its digits equals 9 and when the digits are reversed, the resulting number is 9 greater than the original number. We identified the system of equations that models this scenario and solved for the values of t and u. In this article, we will answer some of the most frequently asked questions about this problem.
Q: What is the two-digit number that satisfies the given conditions?
A: The two-digit number that satisfies the given conditions is 45.
Q: How did you find the values of t and u?
A: We used the method of substitution and elimination to solve the system of equations. We first simplified the system of equations by dividing both sides of the second equation by -9. Then, we added the two equations together to eliminate the variable u. Finally, we substituted the value of t into one of the original equations to find the value of u.
Q: What is the significance of the sum of the digits being 9?
A: The sum of the digits being 9 is a key condition in this problem. It means that the two-digit number must have a certain balance between its tens and units digits.
Q: What is the significance of the new number being 9 greater than the original number?
A: The new number being 9 greater than the original number is another key condition in this problem. It means that when the digits are reversed, the resulting number must be 9 greater than the original number.
Q: Can you explain the concept of a system of equations?
A: A system of equations is a set of two or more equations that are related to each other. In this problem, we had two equations: t + u = 9 and 10u + t = 10t + u + 9. We used these equations to solve for the values of t and u.
Q: How do you know that the solution is unique?
A: We can verify that the solution is unique by plugging the values of t and u back into the original equations. If the equations hold true, then we know that the solution is unique.
Q: Can you provide more examples of two-digit numbers that satisfy the given conditions?
A: Unfortunately, it is not possible to find another two-digit number that satisfies the given conditions. The solution we found is unique.
Q: What is the next step in solving this problem?
A: The next step in solving this problem would be to explore other mathematical concepts that are related to this problem. For example, we could investigate the properties of two-digit numbers and how they relate to the given conditions.
Conclusion
In this article, we answered some of the most frequently asked questions about the problem of finding a two-digit number where the sum of its digits equals 9 and when the digits are reversed, the resulting number is 9 greater than the original number. We hope that this article has provided a deeper understanding of this problem and its solution.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Mathematics for the Nonmathematician" by Morris Kline
Appendix
The following is a list of additional resources that may be helpful in understanding this problem:
- [1] "Introduction to Algebra" by Richard Rusczyk
- [2] "Mathematics for the Modern World" by John Stillwell
By exploring these resources, you can gain a deeper understanding of the mathematical concepts that underlie this problem.