In the realm of mathematics, number sentences serve as a powerful tool for representing and solving problems. These sentences translate real-world scenarios into mathematical equations, allowing us to unravel the mysteries of unknown numbers. This article delves into Exercise 14.1, where we will dissect problems, construct corresponding number sentences, and employ algebraic techniques to determine the values of these elusive unknowns. This exploration is fundamental in developing a strong understanding of algebraic concepts and their applications in everyday problem-solving. Masterfully translating word problems into equations is a key skill in mathematics, enabling us to move beyond rote memorization and engage with the underlying logic and structure of numerical relationships. By focusing on the process of building number sentences, we are able to foster critical thinking and enhance our ability to approach mathematical challenges with confidence and clarity. Each problem in this exercise provides an opportunity to practice this translation process and hone our skills in algebraic manipulation, thereby solidifying our understanding of fundamental mathematical principles. As we work through each scenario, we will not only discover the numerical answers but also develop a deeper appreciation for the elegant way in which mathematics provides a framework for understanding and solving real-world problems.
Problem 1(a): The Addition Mystery
a) If 12 is added to a certain number, the answer is 35.
To tackle this problem, our primary goal is to translate the given word statement into a mathematical equation. This process involves identifying the unknown number, representing it with a variable, and expressing the relationship described in the statement using mathematical symbols. By carefully dissecting the words, we can construct a number sentence that accurately reflects the problem's core concept.
Constructing the Number Sentence:
The phrase "a certain number" immediately signals the presence of an unknown quantity. We can represent this unknown number using a variable, such as x. The statement then tells us that 12 is added to this number, which translates to the mathematical operation of addition (+). Finally, the phrase "the answer is 35" indicates that the result of this addition is equal to 35. Putting it all together, we can write the number sentence as:
x + 12 = 35
This equation encapsulates the essence of the problem, providing a concise and symbolic representation of the given information. It now sets the stage for solving for the unknown, x. This translation process is a crucial step in solving word problems, as it bridges the gap between verbal descriptions and mathematical expressions.
Solving for the Unknown:
Now that we have the number sentence, we can employ algebraic techniques to isolate the variable x and determine its value. The key here is to perform the same operation on both sides of the equation to maintain the equality. In this case, we need to undo the addition of 12. We can achieve this by subtracting 12 from both sides of the equation:
x + 12 - 12 = 35 - 12
This simplifies to:
x = 23
Therefore, the unknown number is 23. This solution represents the value of x that satisfies the original equation, and thus answers the question posed in the word problem. The process of isolating the variable demonstrates a fundamental principle of algebra: the manipulation of equations to reveal the value of an unknown quantity.
Checking the Solution:
To ensure the accuracy of our solution, it's crucial to check our answer. We can do this by substituting the value we found for x (which is 23) back into the original number sentence:
23 + 12 = 35
Performing the addition, we get:
35 = 35
Since this statement is true, our solution is correct. This step of verification is an essential part of the problem-solving process, providing confidence in the accuracy of our results. Checking our solution not only confirms our answer but also reinforces our understanding of the relationship between the variables and the equation.
Problem 1(b): The Subtraction Puzzle
b) If 45 is subtracted from a certain number, the answer is 17.
This problem, similar to the first, presents a scenario involving an unknown number, but this time the operation involved is subtraction. The challenge lies in accurately interpreting the wording and constructing the corresponding mathematical equation. This translation from words to symbols is a cornerstone of mathematical problem-solving, requiring careful attention to detail and an understanding of mathematical operations.
Constructing the Number Sentence:
As before, the phrase "a certain number" indicates the presence of an unknown quantity, which we can represent by the variable y. The key phrase here is "45 is subtracted from a certain number." This means that 45 is being taken away from the unknown number y. In mathematical terms, this translates to y - 45. The statement then tells us that "the answer is 17," which signifies that the result of the subtraction is equal to 17. Combining these elements, we can formulate the number sentence:
y - 45 = 17
This equation accurately represents the relationship described in the word problem. The correct order of operations is crucial in this case, as subtracting 45 from y is different from subtracting y from 45. The number sentence serves as the foundation for solving for the unknown, providing a clear and concise mathematical representation of the problem.
Solving for the Unknown:
To solve for y, we need to isolate the variable on one side of the equation. Since 45 is being subtracted from y, we can undo this operation by adding 45 to both sides of the equation:
y - 45 + 45 = 17 + 45
This simplifies to:
y = 62
Therefore, the unknown number is 62. This result represents the value of y that satisfies the original equation and resolves the subtraction puzzle presented in the word problem. The process of adding 45 to both sides demonstrates a core principle of algebraic manipulation: maintaining balance in the equation to isolate the variable of interest.
Checking the Solution:
To verify the accuracy of our solution, we substitute the value we found for y (which is 62) back into the original number sentence:
62 - 45 = 17
Performing the subtraction, we get:
17 = 17
This statement is true, confirming that our solution is correct. This step of checking our work is vital in ensuring the reliability of our answer and reinforces our understanding of the problem-solving process. It demonstrates the importance of verifying our results to build confidence in our mathematical abilities. By substituting the solution back into the original equation, we are essentially reversing the process to ensure that the equation holds true.
Mastering Number Sentences: A Key to Mathematical Proficiency
These exercises highlight the fundamental skill of translating word problems into mathematical number sentences. This ability forms the bedrock of algebraic problem-solving and is crucial for success in more advanced mathematical concepts. By understanding how to represent unknown quantities with variables and express relationships with mathematical operations, we empower ourselves to tackle a wide range of problems with clarity and confidence. The practice of checking solutions further reinforces this understanding and cultivates a rigorous approach to problem-solving.