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Chapter 2: Problem 75

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. Solving an equation reminds me of keeping a barbell balanced: If I add weight to or subtract weight from one side of the bar, I must do the same thing to the other side.

Short Answer

Expert verified
The statement makes sense, it correctly encapsulates the concept of balancing both sides of an equation by applying the same operations to each.

Step by step solution

01

Understand the statement

The first step is understanding the context of both sides of the analogy: solving equations and balancing a barbell. In mathematical terms, solving an equation is all about keeping the two sides equal, so any operation performed on one side of the equation has to be done to the other side as well. In physical terms, keeping a barbell balanced means maintaining an equal weight distribution, so if weight is added or subtracted from one side, the same must be done to the other side to maintain balance.
02

Analyze the analogy

Next, compare the two processes mentioned in the statement. Both involve maintaining a balance or equality by performing the same operations on both sides. In both scenarios, any change to one side directly impacts the other, requiring a corresponding change to maintain the balance or equality. This implies that the statement captures a fundamental concept of solving equations - whatever is done to one side must also be done to the other to keep the equation true.
03

Make a decision

Based on the analysis in the previous steps, it is clear that the analogy in the statement correctly represents the principles of solving equations. Thus, the statement 'makes sense'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Balance Concept
Picture an equation as a seesaw in a playground, perfectly balanced with equal weights on each side. That's the essence of the equation balance concept. It is a visualization that helps us understand that an equation represents two expressions which are equal to each other. So, when you're working with equations, any changes made to one side must be mirrored on the other.

For example, if you add three to one side of the equation, you must add three to the other side to maintain the balance. This ensures the two sides remain equal, just like the balance of weights would on a seesaw. Understanding this concept helps in all further operations in algebra, such as solving for unknown variables where maintaining this balance is crucial.
Equality in Algebra
At its heart, algebra is about understanding and maintaining equality. When we look at an equation, we鈥檙e actually looking at a statement of equality: the left side is equal to the right side, symbolized by the equal sign (\( = \)). In algebra, when we alter one side of this equation, the principle of equality demands that we carry out the same operation on the opposite side.

This mathematical instruction is not merely a suggestion 鈥� it is a rule to live by if we want to solve equations correctly. Whether we're dealing with addition, subtraction, multiplication, or division, applying operations equally keeps our equation valid and paves the way for finding the solution to our algebraic problem.
Mathematical Analogies
Mathematical analogies are tools that translate complex concepts into relatable or visual scenarios, making them easier to comprehend. Just as the barbell analogy simplifies understanding of the equation balance concept, there are many other analogies we use in mathematics regularly. For instance, comparing the function of parentheses in mathematics to holding a group of items together in a shopping bag helps visualize the concept of grouping in operations.

Another example is likening the process of solving for a variable to finding a missing puzzle piece. These analogies empower students to use their real-world intuition to grasp abstract mathematical principles. By creating a bridge between familiar experiences and new concepts, analogies serve as a meaningful educational tool in mathematics.

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Most popular questions from this chapter

A repair bill on a sailboat came to 1603, dollars including 532 dollars for parts and the remainder for labor. If the cost of labor is 63 dollars per hour, how many hours of labor did it take to repair the sailboat?

Is \(x-4 y=14\) a true statement for \(x=2\) and \(y=-3 ?\)

The bar graph shows average yearly earnings in the United States for people with a college education, by final degree earned. Exercises \(23-24\) are based on the data displayed by the graph. (GRAPH CANNOT COPY) The average yearly salary of an American whose final degree is a doctorate is 39 dollars thousand less than twice that of an American whose final degree is a bachelor's. Combined, two people with each of these educational attainments earn 126 dollars thousand. Find the average yearly salary of Americans with each of these final degrees.

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Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. A number increased by 5 is two times the number. Find the number.
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