Solving Math Problems Finding The Number
In the realm of mathematics, we often encounter intriguing word problems that challenge our analytical and problem-solving skills. These problems require us to translate verbal descriptions into mathematical equations and then manipulate those equations to arrive at a solution. One such captivating problem involves unraveling the identity of a mystery number based on a given relationship. In this comprehensive guide, we will embark on a step-by-step journey to dissect this problem, formulate the appropriate equation, and ultimately unveil the hidden number.
At the heart of any word problem lies a narrative, a story woven with mathematical clues. Our first task is to carefully decipher this narrative, extracting the essential information that will guide us towards the solution. In this particular problem, we are presented with a statement that describes a relationship between an unknown number and the operations of subtraction, multiplication, and addition. The statement reads: "The difference of a number and 6 is the same as 5 times the sum of the number and 2." To truly grasp the essence of this statement, we need to break it down into smaller, more digestible components.
The phrase "the difference of a number and 6" implies that we are subtracting 6 from an unknown number. Let's represent this unknown number with the variable x. Therefore, "the difference of a number and 6" can be expressed mathematically as x - 6. Next, we encounter the phrase "is the same as," which signifies an equality. In mathematical terms, this translates to the equals sign (=). Moving forward, we have "5 times the sum of the number and 2." This indicates that we are multiplying 5 by the result of adding the unknown number (x) and 2. The sum of the number and 2 can be written as x + 2, and multiplying this sum by 5 gives us 5(x + 2). Now, we have successfully deconstructed the problem statement into its mathematical building blocks. We have identified the operations involved, the relationships between the quantities, and the key phrases that translate into mathematical symbols. With this understanding, we are well-equipped to construct the equation that will lead us to the solution.
With the problem statement thoroughly dissected, we can now embark on the crucial step of translating the verbal description into a mathematical equation. This equation will serve as the foundation for our solution, providing a symbolic representation of the relationships described in the problem. Recall that we identified the following components:
- "The difference of a number and 6": x - 6
- "is the same as": =
- "5 times the sum of the number and 2": 5(x + 2)
By combining these components, we can construct the equation:
x - 6 = 5(x + 2)
This equation encapsulates the essence of the problem statement, expressing the equality between the difference of the number and 6 and 5 times the sum of the number and 2. The variable x represents the unknown number that we seek to uncover. With the equation in place, we can now proceed to the next stage: solving for x.
Now comes the exciting part – the process of unraveling the equation to reveal the value of x, our mystery number. This involves applying algebraic techniques to isolate x on one side of the equation. Let's break down the solution step-by-step:
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Distribute: Our first task is to simplify the right side of the equation by distributing the 5 across the terms inside the parentheses. This means multiplying 5 by both x and 2:
x - 6 = 5x + 10
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Isolate x terms: Next, we want to gather all the terms containing x on one side of the equation. To achieve this, we can subtract x from both sides:
x - 6 - x = 5x + 10 - x
This simplifies to:
-6 = 4x + 10
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Isolate constant terms: Now, we need to isolate the constant terms (the numbers without x) on the other side of the equation. We can accomplish this by subtracting 10 from both sides:
-6 - 10 = 4x + 10 - 10
This simplifies to:
-16 = 4x
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Solve for x: Finally, to isolate x, we divide both sides of the equation by 4:
-16 / 4 = 4x / 4
This gives us:
-4 = x
Therefore, the solution to the equation is x = -4. This means that the mystery number we were seeking is -4.
In mathematics, it's always a good practice to verify our solutions to ensure accuracy. To do this, we substitute the value we found for x back into the original equation and check if both sides of the equation are equal. Our original equation was:
x - 6 = 5(x + 2)
Substituting x = -4, we get:
-4 - 6 = 5(-4 + 2)
Simplifying both sides:
-10 = 5(-2)
-10 = -10
Since both sides of the equation are equal, our solution x = -4 is verified. We have successfully solved the equation and confirmed that -4 is indeed the mystery number.
Having diligently worked through the problem, we arrive at the answer. The number that satisfies the given conditions is -4. Therefore, the correct answer is A. -4.
In this comprehensive guide, we embarked on a journey to solve a captivating word problem. We learned how to dissect the problem statement, translate it into a mathematical equation, and then apply algebraic techniques to solve for the unknown number. We also emphasized the importance of verifying our solutions to ensure accuracy. This problem serves as a valuable illustration of the power of mathematics in unraveling real-world puzzles. By mastering these problem-solving skills, we can confidently tackle a wide range of mathematical challenges.
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- Word problems
- Algebraic equations
- Solving equations
- Linear equations
- Problem-solving
- Mathematical reasoning
- Translating words to equations
- Verification of solutions
Understanding the Question
Before diving into the solution, let's first make sure we fully understand the question being asked. The core of the problem lies in the statement: "The difference of a number and 6 is the same as 5 times the sum of the number and 2." Our goal is to identify the specific number that satisfies this condition. To achieve this, we'll need to translate this verbal statement into a mathematical equation, a process that involves representing the unknown number with a variable and expressing the relationships between the numbers and operations using mathematical symbols. This initial step is crucial as it lays the foundation for our subsequent calculations and ultimately leads us to the correct answer. By carefully analyzing the wording of the problem and identifying the key components, we can effectively bridge the gap between the verbal description and its mathematical representation.
Setting up the Equation
Now, let's embark on the process of transforming the verbal statement into a mathematical equation. This involves representing the unknown number with a variable, let's use x, and then expressing the relationships between the numbers and operations using mathematical symbols. The phrase "the difference of a number and 6" translates to x - 6. The phrase "is the same as" signifies equality, represented by the equals sign (=). Finally, "5 times the sum of the number and 2" can be expressed as 5(x + 2). By combining these components, we arrive at the equation:
x - 6 = 5(x + 2)
This equation forms the bedrock of our solution, encapsulating the essence of the problem statement in a concise and mathematically actionable form. With the equation established, we are now poised to move forward and solve for the unknown variable x, which represents the number we are seeking. The subsequent steps will involve applying algebraic techniques to isolate x and ultimately determine its value.
Solving for the Unknown
With the equation x - 6 = 5(x + 2) firmly in place, our next step is to solve for the unknown variable x. This involves employing algebraic manipulation techniques to isolate x on one side of the equation. First, we distribute the 5 on the right side: x - 6 = 5x + 10. Next, we want to gather all the terms containing x on one side of the equation. To achieve this, we can subtract x from both sides, resulting in -6 = 4x + 10. Now, we isolate the constant terms by subtracting 10 from both sides, giving us -16 = 4x. Finally, to solve for x, we divide both sides by 4, which yields x = -4. Therefore, the solution to the equation, and consequently the answer to our problem, is x = -4. This value represents the number that satisfies the given conditions, making it the key to unlocking the puzzle presented in the problem statement.
Checking the Solution
In the realm of mathematics, it's always a prudent practice to verify our solutions to ensure accuracy and to catch any potential errors that may have crept into our calculations. To do this, we substitute the value we found for x, which is -4, back into the original equation and check if both sides of the equation are indeed equal. Our original equation was x - 6 = 5(x + 2). Substituting x = -4, we get -4 - 6 = 5(-4 + 2). Simplifying both sides, we have -10 = 5(-2), which further simplifies to -10 = -10. Since both sides of the equation are equal, our solution x = -4 is verified. This confirmation step provides us with the assurance that our calculations were correct and that we have indeed found the accurate solution to the problem. The verification process not only reinforces our confidence in the answer but also serves as a valuable learning tool, solidifying our understanding of the problem-solving process.
The Correct Choice
Having diligently worked through the problem, meticulously setting up the equation, solving for the unknown, and verifying our solution, we can confidently arrive at the answer. The number that satisfies the given conditions is -4. Therefore, the correct answer among the options provided is:
A. -4
This concludes our step-by-step journey through the problem, demonstrating how careful analysis, algebraic techniques, and verification can lead us to the correct solution. By mastering these skills, we can approach similar mathematical challenges with confidence and precision.
