Solving The Equation R + 14 = 39 A Step-by-Step Guide
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In the realm of mathematics, equations serve as fundamental tools for expressing relationships between quantities and solving for unknown values. This article delves into the equation R + 14 = 39, a simple yet illustrative example of an algebraic equation. We will embark on a step-by-step journey to unravel the mystery behind this equation, understand its components, and ultimately determine the value of the unknown variable, R. This exploration is designed not just to solve this particular problem but also to illuminate the broader principles of algebraic problem-solving, making it accessible and engaging for learners of all levels. This introductory section sets the stage for a detailed analysis, ensuring that readers grasp the core concepts before diving into the solution process. By focusing on clarity and comprehension, we aim to empower individuals with the skills and confidence to tackle similar mathematical challenges.
Dissecting the Equation: R + 14 = 39
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At its heart, the equation R + 14 = 39 is a concise mathematical statement asserting the equality between two expressions. On the left-hand side, we encounter the expression R + 14, where R represents the unknown variable we seek to find. The number 14 is a constant, a fixed value that remains unchanged throughout the equation. The plus sign (+) signifies addition, indicating that R and 14 are to be combined. On the right-hand side, we have the constant 39, which represents the total value or result of the addition performed on the left-hand side. The equals sign (=) is the linchpin of the equation, firmly establishing the balance between the two sides. It conveys that the value of the expression R + 14 is precisely the same as the value of 39. Understanding these components – the variable, constants, and the equals sign – is crucial for comprehending the equation's meaning and devising a strategy for solving it. This section lays the groundwork for the subsequent steps, ensuring that readers have a solid grasp of the equation's structure and the relationships it embodies. The key here is to recognize the equation as a balanced statement, where our goal is to isolate R and discover its value.
The Goal: Isolating the Variable R
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The primary objective in solving any algebraic equation is to isolate the variable, in this case, R. Isolating R means manipulating the equation in such a way that R stands alone on one side of the equals sign, with its value clearly revealed on the other side. This process involves strategically applying mathematical operations to both sides of the equation, ensuring that the balance is maintained. Think of the equation as a weighing scale; to keep it balanced, any operation performed on one side must be mirrored on the other. In the equation R + 14 = 39, R is currently accompanied by the constant 14, which is being added to it. To isolate R, we need to undo this addition. The inverse operation of addition is subtraction, so we will subtract 14 from both sides of the equation. This section underscores the importance of understanding inverse operations in algebra, setting the stage for the actual solution process. By emphasizing the concept of maintaining balance, we reinforce a fundamental principle of equation solving. The anticipation of the next step – subtracting 14 from both sides – is built upon a solid understanding of the goal and the means to achieve it.
Applying the Subtraction Property of Equality
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The subtraction property of equality is a cornerstone principle in algebra, asserting that subtracting the same value from both sides of an equation preserves the equality. This principle is precisely what we need to isolate R in the equation R + 14 = 39. To apply this property, we subtract 14 from both sides of the equation. This can be represented mathematically as follows:
R + 14 - 14 = 39 - 14
On the left-hand side, subtracting 14 effectively cancels out the addition of 14, leaving R by itself. On the right-hand side, we perform the subtraction 39 - 14, which yields 25. This section provides a clear and concise explanation of the subtraction property of equality, a critical concept for solving algebraic equations. The step-by-step application of this property to the given equation is demonstrated, making the process transparent and easy to follow. The emphasis on maintaining equality reinforces the importance of balanced operations in algebraic manipulations. The transition from theory to practice is seamless, bridging the gap between understanding the principle and applying it effectively.
Simplifying the Equation
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After applying the subtraction property, we simplify both sides of the equation to reveal the value of R. On the left-hand side, R + 14 - 14 simplifies to just R, as the +14 and -14 cancel each other out. This is a fundamental arithmetic operation that streamlines the equation and brings us closer to our goal. On the right-hand side, the subtraction 39 - 14 results in 25. Therefore, the equation simplifies to:
R = 25
This section highlights the simplification process, a crucial step in solving equations. The clear demonstration of how the left-hand side reduces to R emphasizes the effectiveness of the subtraction property. The arithmetic calculation on the right-hand side is straightforward, ensuring that readers can easily follow the steps. The resulting equation, R = 25, is a concise and definitive statement of the solution. This section underscores the power of simplification in revealing the value of the unknown variable.
The Solution: R = 25
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The simplified equation R = 25 provides the solution to the original problem. It unequivocally states that the value of the unknown variable R is 25. This means that if we substitute 25 for R in the original equation, R + 14 = 39, the equation holds true. We can verify this by performing the addition:
25 + 14 = 39
This section presents the solution in a clear and concise manner, emphasizing its significance. The verification step reinforces the correctness of the solution and provides a sense of closure. The connection back to the original equation underscores the importance of the solution in the context of the problem. The confidence in the answer is bolstered by the successful verification, highlighting the accuracy of the entire solution process. This section serves as a culmination of the previous steps, solidifying the understanding of how to solve the equation and interpret the result.
Verifying the Solution: A Crucial Step
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Verifying the solution is a critical step in the problem-solving process, ensuring the accuracy of our result. To verify that R = 25 is indeed the correct solution, we substitute 25 for R in the original equation, R + 14 = 39. This substitution transforms the equation into a numerical statement:
25 + 14 = 39
Now, we perform the addition on the left-hand side:
39 = 39
Since the left-hand side equals the right-hand side, the equation holds true. This confirms that our solution, R = 25, is correct. Verification is not just a formality; it's a safeguard against errors and a way to build confidence in our problem-solving abilities. This section emphasizes the importance of verification as an integral part of the solution process. The step-by-step demonstration of the verification process reinforces the concept and provides a clear example for readers to follow. The emphasis on accuracy and confidence building highlights the value of this crucial step.
Understanding the Answer Choices
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In the context of multiple-choice questions, understanding the answer choices is crucial for selecting the correct one. The given answer choices for the equation R + 14 = 39 are:
A. 24 B. 546 C. 53 D. 25
We have already determined that the correct value of R is 25. Therefore, the correct answer choice is D. It's important to note how incorrect answer choices can arise from common mistakes in algebraic problem-solving. For instance, a student might mistakenly add 14 to 39 instead of subtracting, leading to answer choice C (53). Similarly, other errors in arithmetic or algebraic manipulation could lead to the other incorrect choices. This section contextualizes the solution within the framework of multiple-choice questions, emphasizing the importance of understanding the answer choices. The analysis of potential errors that could lead to incorrect choices provides valuable insights for learners. The connection between the calculated solution and the correct answer choice reinforces the understanding of the problem-solving process.
Why Other Options Are Incorrect
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To fully grasp the solution, it's beneficial to understand why the other answer choices are incorrect. Let's examine each option:
A. 24: If we substitute 24 for R in the original equation, we get 24 + 14 = 38, which is not equal to 39. Therefore, 24 is not the correct value of R. B. 546: This value is significantly larger than 39 and could result from a misunderstanding of the equation or an incorrect operation. Substituting 546 for R would result in a much larger sum than 39. C. 53: As mentioned earlier, this answer could arise from mistakenly adding 14 to 39 instead of subtracting. Substituting 53 for R gives 53 + 14 = 67, which is not equal to 39.
By understanding why these options are incorrect, we reinforce our understanding of the correct solution and the problem-solving process. This section provides a comprehensive analysis of the incorrect answer choices, enhancing the understanding of the solution process. The explanations of potential errors that could lead to these choices serve as valuable learning points. The reinforcement of the correct solution through the elimination of incorrect options solidifies the comprehension of the problem.
Conclusion: Mastering Algebraic Equations
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In conclusion, solving the equation R + 14 = 39 demonstrates the fundamental principles of algebraic problem-solving. By understanding the equation's components, applying the subtraction property of equality, simplifying the equation, and verifying the solution, we successfully determined that R = 25. This process not only provides the answer to this specific problem but also equips us with the skills to tackle a wide range of algebraic challenges. The ability to isolate variables and solve equations is a cornerstone of mathematical proficiency, essential for further studies in mathematics and various scientific disciplines. This article has aimed to demystify the process, making it accessible and engaging for learners of all backgrounds. By focusing on clarity, step-by-step explanations, and the importance of verification, we hope to empower readers with the confidence and competence to excel in algebra. The journey through this equation serves as a microcosm of the broader world of mathematical problem-solving, where logical reasoning, careful manipulation, and a commitment to accuracy are key to success.
