Solving The Equation The Sum Of Eight And Three Times A Number Is 92

In the realm of mathematics, we often encounter intriguing problems that challenge our analytical skills and problem-solving abilities. One such problem involves deciphering the relationship between numbers and translating them into mathematical equations. Let's delve into a specific scenario: "The sum of eight and three times a number is 92. What is the number?"

This seemingly simple statement holds the key to unlocking a numerical mystery. To unravel this puzzle, we must first transform the words into a symbolic representation – a mathematical equation. This equation will serve as a roadmap, guiding us through the steps necessary to pinpoint the elusive number.

A. Translating Words into Equations: The Foundation of Mathematical Problem-Solving

Our primary task is to convert the given statement into a precise equation. This involves identifying the unknown quantity, representing it with a variable, and then expressing the relationships described in the statement using mathematical symbols.

In this case, the unknown quantity is "the number" we are trying to find. Let's represent this number with the variable 'x'. Now, let's break down the statement piece by piece and translate it into mathematical expressions.

The statement begins with "the sum of eight and..." This indicates that we are adding eight to something. The next part of the statement is "three times a number." This means we are multiplying the number (which we've represented as 'x') by three. So, "three times a number" can be written as 3x.

Putting these pieces together, "the sum of eight and three times a number" can be expressed as 8 + 3x. The statement continues with "is 92," which simply means equals 92. Therefore, we can write the entire statement as the equation:

8 + 3x = 92

This equation is the mathematical representation of the problem, and it is the foundation upon which we will build our solution. It encapsulates the relationships between the known quantities (8 and 92) and the unknown quantity (x).

Understanding the Importance of Accurate Translation

The accuracy of this translation is paramount. A slight error in converting the words into symbols can lead to an entirely different equation, and consequently, an incorrect solution. Therefore, it's crucial to meticulously dissect the statement and ensure that each element is accurately represented in the equation.

For instance, if we misinterpreted "three times a number" as x + 3 instead of 3x, the equation would be drastically different, and the solution we obtain would be wrong. Similarly, if we overlooked the "sum of" part and simply wrote 8 * 3x = 92, we would be deviating from the problem's intended meaning.

Therefore, the process of translating words into equations requires careful attention to detail and a thorough understanding of mathematical terminology and symbolism. It is a fundamental skill in problem-solving, not just in mathematics, but also in various fields like physics, engineering, and economics.

In the subsequent sections, we will delve into the techniques for solving this equation, uncovering the value of 'x', which represents the number we are seeking. However, it's crucial to remember that the correctness of the solution hinges on the accuracy of this initial translation.

B. Solving the Equation: Unraveling the Numerical Mystery

Now that we have successfully translated the problem into the equation 8 + 3x = 92, we can proceed to the next crucial step: solving the equation for 'x'. Solving an equation involves isolating the variable on one side of the equation to determine its value. We achieve this by performing algebraic operations on both sides of the equation, maintaining the balance and equality throughout the process.

Step 1: Isolating the Term with the Variable

Our first goal is to isolate the term containing the variable 'x', which is 3x. To do this, we need to eliminate the constant term, 8, from the left side of the equation. We can accomplish this by subtracting 8 from both sides of the equation:

8 + 3x - 8 = 92 - 8

This simplifies to:

3x = 84

By subtracting 8 from both sides, we have effectively isolated the term 3x on the left side, bringing us closer to our goal of determining the value of 'x'.

Step 2: Solving for the Variable

Now that we have 3x = 84, we need to isolate 'x' completely. To do this, we need to undo the multiplication by 3. We achieve this by dividing both sides of the equation by 3:

3x / 3 = 84 / 3

This simplifies to:

x = 28

Therefore, we have successfully solved the equation and found the value of 'x', which is 28. This means that the number we were seeking is 28.

Verifying the Solution

It's always a good practice to verify our solution to ensure its accuracy. We can do this by substituting the value of 'x' back into the original equation and checking if it holds true.

Substituting x = 28 into the equation 8 + 3x = 92, we get:

8 + 3(28) = 92

8 + 84 = 92

92 = 92

Since the equation holds true, our solution x = 28 is correct. This confirms that the number we were looking for is indeed 28.

Understanding the Principles of Equation Solving

The process of solving equations relies on fundamental algebraic principles. The key principle is maintaining the equality of the equation. Any operation performed on one side of the equation must also be performed on the other side to preserve the balance.

For example, when we subtracted 8 from both sides of the equation, we ensured that the equation remained balanced. Similarly, when we divided both sides by 3, we maintained the equality. If we had performed the operation on only one side, the equation would no longer be valid.

Another important principle is the concept of inverse operations. To isolate a variable, we use the inverse operation of the operation being performed on it. For example, the inverse operation of addition is subtraction, and the inverse operation of multiplication is division. By applying inverse operations, we can systematically undo the operations surrounding the variable and isolate it.

The ability to solve equations is a cornerstone of mathematical problem-solving. It is a skill that is widely applicable in various fields, including science, engineering, finance, and computer science. Mastering equation-solving techniques empowers us to tackle complex problems and make informed decisions.

In conclusion, by meticulously translating the given statement into an equation and applying the principles of equation solving, we have successfully determined the number to be 28. This exercise highlights the power of mathematics in unraveling numerical mysteries and provides a foundation for tackling more complex problems in the future.