Solving For Three Numbers Given Product And Difference Relationships
Introduction
In this mathematical exploration, we delve into the intriguing problem of finding three specific numbers based on the relationships between their products and differences. This task involves a blend of algebraic manipulation and logical deduction, providing a fascinating challenge for math enthusiasts. The core of the problem lies in deciphering the given clues: the product of the first two numbers, the product of the first and third numbers, and the difference between the last two. By carefully analyzing these relationships, we can construct a system of equations that will lead us to the solution. The beauty of this problem is that it showcases how seemingly simple mathematical principles can be applied to unravel complex scenarios. Let's embark on this journey of discovery and unravel the mystery of these three elusive numbers.
Problem Statement
Unlocking the Numerical Puzzle: Imagine you are presented with a numerical puzzle where you need to identify three unknown numbers. You are given three crucial pieces of information. First, the product of the first two numbers is 270. Second, the product of the first and third numbers is 150. Third, the difference between the last two numbers is 8. The challenge is to use these clues to determine the values of the three numbers. This is a classic mathematical problem that requires a systematic approach to solve. To begin, we need to translate these statements into algebraic equations. Let's denote the three numbers as x, y, and z. The given information can then be written as follows:
- x * y = 270
- x * z = 150
- z - y = 8
These three equations form a system that we can solve to find the values of x, y, and z. The next step involves choosing a strategy to solve this system. We can use substitution, elimination, or a combination of both. Each method has its advantages, and the choice often depends on the specific structure of the equations. In this case, we can start by expressing one variable in terms of another and then substituting it into the other equations. This will help us reduce the number of unknowns and simplify the system. Let's proceed with this approach and see where it leads us.
Setting up the Equations
Translating Words into Math: To solve this numerical puzzle, our initial step involves transforming the given information into a set of algebraic equations. This is a crucial step in any mathematical problem-solving process, as it allows us to manipulate the relationships between the unknowns in a precise and systematic way. Let's denote the three numbers as x, y, and z. With these variables in place, we can now express the given conditions as follows. The product of the first two numbers is 270, which translates to the equation:
x * y = 270
This equation establishes a direct relationship between x and y, indicating that their product is a constant value. Next, we are told that the product of the first and third numbers is 150. This gives us the equation:
x * z = 150
Similar to the first equation, this one links x and z through their product. Finally, the difference between the last two numbers is 8, which can be written as:
z - y = 8
This equation introduces a linear relationship between y and z, stating that their difference is a fixed value. Now that we have these three equations, we can see how they form a system that can be solved to find the values of x, y, and z. The next phase in our solution involves strategically manipulating these equations to isolate the variables and determine their values. This might involve substitution, elimination, or other algebraic techniques. The key is to choose a method that simplifies the system and allows us to solve for the unknowns efficiently. Let's explore the possible approaches and decide on the most effective strategy.
Solving the System of Equations
Unraveling the Unknowns: With our equations neatly set up, the next crucial phase is to solve the system and unveil the values of our three mysterious numbers. We have the following equations at our disposal:
- x * y* = 270
- x * z* = 150
- z - y = 8
A strategic approach here is to use a combination of substitution and algebraic manipulation. Let's begin by isolating x in both the first and second equations. From the first equation, we have:
x = 270 / y
And from the second equation, we get:
x = 150 / z
Since both expressions are equal to x, we can set them equal to each other:
270 / y = 150 / z
This gives us a new equation that relates y and z. Now, let's cross-multiply to simplify:
270 * z = 150 * y
We can further simplify this by dividing both sides by 30:
9 * z = 5 * y
This equation, along with our third original equation (z - y = 8), forms a new system of two equations with two variables (y and z). We can now use substitution or elimination to solve this system. Let's express z in terms of y from the third original equation:
z = y + 8
Now, substitute this into our simplified equation:
9 * (y + 8) = 5 * y
This equation is now in terms of y only, making it solvable. By expanding and rearranging, we can find the value of y. Once we have y, we can easily find z using z = y + 8. Finally, we can substitute the values of y and z back into the original equations to find x. This methodical approach will lead us to the solution, one variable at a time.
Finding the Values of x, y, and z
The Grand Finale of Calculation: Now we're on the verge of solving the puzzle. Let's take the equation we derived:
9 * (y + 8) = 5 * y
Expand the left side:
9y + 72 = 5y
Now, let's bring the y terms to one side and the constants to the other:
9y - 5y = -72
Simplify:
4y = -72
Divide by 4 to solve for y:
y = -18
Great! We've found y. Now we can use the equation z = y + 8 to find z:
z = -18 + 8
z = -10
So, we have y = -18 and z = -10. The final step is to find x. We can use either x * y* = 270 or x * z* = 150. Let's use the first one:
x * (-18) = 270
Divide by -18 to solve for x:
x = 270 / (-18)
x = -15
Thus, we've found all three numbers: x = -15, y = -18, and z = -10. These numbers satisfy all the given conditions: the product of the first two is 270, the product of the first and third is 150, and the difference between the last two is 8. This methodical solution demonstrates the power of algebraic manipulation in solving complex problems.
Verification of the Solution
Ensuring Accuracy: Before we celebrate our victory, it's crucial to verify that our solution is indeed correct. This step ensures that the values we've found for x, y, and z satisfy all the original conditions given in the problem. We found that x = -15, y = -18, and z = -10. Let's plug these values back into our original equations and see if they hold true. The first condition was that the product of the first two numbers (x and y) is 270. So, we need to check if:
x * y = 270
Substituting our values:
(-15) * (-18) = 270
This is indeed true, as -15 multiplied by -18 equals 270. Next, we need to check the second condition: the product of the first and third numbers (x and z) is 150. This means we need to verify:
x * z = 150
Plugging in our values:
(-15) * (-10) = 150
This also holds true, as -15 multiplied by -10 equals 150. Finally, we need to verify the third condition: the difference between the last two numbers (z and y) is 8. So, we check if:
z - y = 8
Substituting our values:
(-10) - (-18) = 8
This is also correct, as -10 minus -18 equals 8. Since all three conditions are satisfied by our values of x, y, and z, we can confidently conclude that our solution is accurate. This verification step underscores the importance of checking our work in mathematical problem-solving to ensure the correctness of our results.
Conclusion
The Triumph of Mathematical Deduction: In this exploration, we successfully navigated a numerical puzzle, demonstrating the power and elegance of mathematical problem-solving. We were tasked with finding three unknown numbers, given the product of the first two (270), the product of the first and third (150), and the difference between the last two (8). By translating these conditions into algebraic equations, we formed a system that could be solved through strategic manipulation. We denoted the three numbers as x, y, and z, and expressed the given information as follows:
- x * y* = 270
- x * z* = 150
- z - y = 8
Our approach involved isolating variables, substituting expressions, and simplifying equations. We first expressed x in terms of y and z, then equated the expressions to create a relationship between y and z. This led us to a new system of two equations with two variables, which we solved using substitution. We found that y = -18 and z = -10. With these values, we could easily find x using one of the original equations, resulting in x = -15. To ensure the accuracy of our solution, we meticulously verified that the values x = -15, y = -18, and z = -10 satisfied all the original conditions. This verification step confirmed the correctness of our results. The journey through this problem highlights the importance of systematic thinking, algebraic manipulation, and verification in mathematical problem-solving. It also showcases how seemingly complex problems can be broken down into smaller, manageable steps, leading to a clear and satisfying solution. The ability to translate real-world scenarios into mathematical models and solve them is a valuable skill that can be applied in various fields. This puzzle serves as a testament to the beauty and utility of mathematics in unraveling the mysteries of numbers and relationships.
