Finding Three Numbers With Given Products And Difference A Step By Step Guide

Introduction

This article delves into a mathematical problem involving the determination of three unknown numbers based on specific relationships between them. We are given the product of the first two numbers, the product of the first and third numbers, and the difference between the last two numbers. Our goal is to find these three numbers using the provided information. This type of problem often involves setting up a system of equations and solving it using algebraic techniques. Understanding how to solve such problems is crucial in developing problem-solving skills and applying mathematical concepts in real-world scenarios. Solving for unknown numbers using given relationships requires careful analysis and a systematic approach. We will explore the steps involved in setting up the equations and solving them to find the three numbers.

Problem Statement

We are tasked with finding three numbers under the following conditions:

1. The product of the first two numbers is 270.
2. The product of the first and third numbers is 150.
3. The difference between the last two numbers is 8.

Let's denote the three numbers as x, y, and z. We can translate the given information into a system of equations:

• Equation 1: x * y* = 270
• Equation 2: x * z* = 150
• Equation 3: z - y = 8

This system of equations forms the foundation for solving our problem. We will employ various algebraic techniques to find the values of x, y, and z that satisfy all three equations. Solving this system will require strategic manipulation and substitution to isolate the variables and determine their values. The challenge lies in effectively combining these equations to eliminate variables and simplify the problem.

Solving the System of Equations

To solve the system of equations, we can use a combination of substitution and algebraic manipulation. Our system of equations is:

1. x * y* = 270
2. x * z* = 150
3. z - y = 8

First, we can express y and z in terms of x using equations 1 and 2:

• From equation 1: y = 270 / x
• From equation 2: z = 150 / x

Now, we substitute these expressions for y and z into equation 3:

(150 / x) - (270 / x) = 8

To simplify this equation, we find a common denominator, which is x:

(150 - 270) / x = 8

-120 / x = 8

Now, we solve for x by multiplying both sides by x and then dividing by 8:

-120 = 8 * x

x = -120 / 8

x = -15

Now that we have the value of x, we can find y and z using the expressions we derived earlier:

• y = 270 / x = 270 / -15 = -18
• z = 150 / x = 150 / -15 = -10

Thus, the three numbers are x = -15, y = -18, and z = -10. We can verify our solution by plugging these values back into the original equations:

• (-15) * (-18) = 270 (Equation 1 is satisfied)
• (-15) * (-10) = 150 (Equation 2 is satisfied)
• (-10) - (-18) = 8 (Equation 3 is satisfied)

Solving for variables in a system of equations often involves strategic manipulation to isolate each variable. In this case, expressing y and z in terms of x allowed us to simplify the problem and solve for x directly. The subsequent steps then involved substituting the value of x back into the equations to find y and z.

Alternative Approach

An alternative approach to solving this system involves manipulating the equations in a slightly different way. Starting with the same system of equations:

1. x * y* = 270
2. x * z* = 150
3. z - y = 8

We can divide Equation 1 by Equation 2 to eliminate x:

(x * y*) / (x * z*) = 270 / 150

y / z = 9 / 5

Now we have a new equation:

4. y = (9/5) * z

Substitute this expression for y into Equation 3:

z - (9/5) * z = 8

To simplify, multiply both sides by 5:

5z - 9z = 40

-4z = 40

z = -10

Now that we have z, we can find y using Equation 4:

y = (9/5) * (-10) = -18

And finally, we can find x using Equation 2:

x * (-10) = 150

x = -15

This alternative method confirms our previous solution: x = -15, y = -18, and z = -10. This approach demonstrates the flexibility in solving systems of equations and highlights how different algebraic manipulations can lead to the same result. Employing different methods to solve the same problem can provide a deeper understanding of the underlying mathematical principles and reinforce problem-solving skills.

Verification and Conclusion

We have found the three numbers to be x = -15, y = -18, and z = -10. To ensure our solution is correct, we need to verify that these values satisfy all three original equations:

1. x * y* = (-15) * (-18) = 270 (Correct)
2. x * z* = (-15) * (-10) = 150 (Correct)
3. z - y = (-10) - (-18) = 8 (Correct)

Since all three equations are satisfied, our solution is verified. In conclusion, the three numbers that satisfy the given conditions are -15, -18, and -10. Verifying the solution is a crucial step in problem-solving as it ensures the accuracy of the results. This problem demonstrates the application of algebraic techniques in solving systems of equations and highlights the importance of careful manipulation and substitution. Understanding these methods is essential for tackling more complex mathematical problems.

Real-World Applications

While this problem may seem purely mathematical, the underlying concepts have applications in various real-world scenarios. Solving systems of equations is a fundamental skill in fields such as engineering, physics, economics, and computer science. For example, in engineering, systems of equations are used to analyze circuits, design structures, and model fluid flow. In economics, they are used to model supply and demand, analyze market equilibrium, and make predictions about economic trends. In computer science, they are used in optimization algorithms, data analysis, and machine learning.

Consider a scenario in economics where you need to determine the equilibrium price and quantity of a product. You might have two equations: one representing the supply curve and another representing the demand curve. Solving this system of equations will give you the equilibrium price and quantity, which are crucial for understanding market dynamics. Similarly, in engineering, you might have a set of equations describing the forces acting on a structure. Solving this system will allow you to determine the stresses and strains in the structure and ensure its stability.

Another application can be seen in computer graphics, where systems of equations are used to transform and manipulate objects in 3D space. Understanding how to solve these systems is essential for creating realistic and interactive graphics. Moreover, in cryptography, solving systems of equations can be used to break certain types of codes, highlighting the importance of mathematical problem-solving in security contexts. The ability to translate real-world problems into mathematical models and solve them using algebraic techniques is a valuable skill in many professional domains.

Tips for Solving Similar Problems

When faced with similar problems involving finding unknown numbers based on given relationships, there are several strategies you can employ to increase your chances of success. Effective problem-solving often involves breaking down the problem into smaller, more manageable parts and systematically addressing each part. Here are some tips to keep in mind:

1. Carefully read and understand the problem statement: Before attempting to solve the problem, make sure you fully understand what is being asked. Identify the unknowns and the relationships between them. Write down the given information clearly and concisely.

2. Translate the problem into a system of equations: Convert the verbal descriptions into mathematical equations. This is a crucial step in solving the problem. Make sure you have as many independent equations as unknowns. If you have fewer equations than unknowns, the problem may have multiple solutions or no solution.

3. Choose an appropriate method for solving the system: There are several methods for solving systems of equations, including substitution, elimination, and matrix methods. Choose the method that seems most appropriate for the given problem. Sometimes, a combination of methods may be necessary.

4. Check your solution: Once you have found a solution, verify that it satisfies all the original equations. This will help you catch any errors you may have made in your calculations. If the solution does not satisfy the equations, go back and review your steps.

5. Look for alternative approaches: If you are stuck on a problem, try a different approach. There may be more than one way to solve the problem. Sometimes, a different perspective can lead to a breakthrough.

6. Practice regularly: The more you practice solving problems, the better you will become at it. Work through a variety of examples and try different types of problems. This will help you develop your problem-solving skills and build your confidence.

By following these tips, you can improve your ability to solve problems involving finding unknown numbers and apply these skills in various contexts. The key is to approach each problem systematically, break it down into manageable steps, and practice consistently.

Conclusion

In summary, we successfully found three numbers given their products and difference by setting up and solving a system of equations. We explored two different methods for solving the system, demonstrating the flexibility and adaptability required in mathematical problem-solving. The solution, x = -15, y = -18, and z = -10, was verified to ensure accuracy. This exercise underscores the importance of algebraic techniques in solving mathematical problems and their applicability in real-world scenarios. Mastering these problem-solving skills is crucial for success in various fields, including engineering, economics, and computer science. The ability to translate complex problems into mathematical models and solve them systematically is a valuable asset in both academic and professional pursuits. By practicing and applying these techniques, individuals can enhance their analytical thinking and problem-solving capabilities.