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Chapter 2: Problem 6

Justine asked her group members to do this calculation: Pick a number, multiply by 5 , and subtract 2. Quentin got 33 for an answer. Explain how Justine could determine what number Quentin picked. What number did Quentin pick?

Short Answer

Expert verified
Quentin picked 7.

Step by step solution

01

Understand the Relationship

Justine's group members were asked to pick a number, which we'll call \( x \). Then they were to multiply it by 5 and subtract 2. Quentin's answer was 33, so we set up the equation: \( 5x - 2 = 33 \).
02

Isolate the Variable

To find \( x \), we need to solve for it. First, add 2 to both sides of the equation to get rid of the \(-2\):\[ 5x - 2 + 2 = 33 + 2 \]This simplifies to:\[ 5x = 35 \]
03

Solve for x

Now, divide both sides of the equation by 5 to solve for \( x \):\[ \frac{5x}{5} = \frac{35}{5} \]This simplifies to:\[ x = 7 \]

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Key Concepts

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

Problem Solving
When faced with an algebraic problem like Justine's, the goal is to find unknowns based on given conditions. In this exercise, Quentin ends up with 33 after picking a number, multiplying it by 5, and subtracting 2. Our task is to reverse-engineer the problem to understand which number, when processed as per the instructions, would yield 33.Key steps in problem solving include:
  • Understanding the problem: Identify what needs to be found and the information given.
  • Setting up equations: Convert the problem's words into mathematical expressions.
  • Planning a solution: Decide on a method to isolate and solve for the unknown variable.
In this scenario, we transform Justine's instructions into the equation: \( 5x - 2 = 33 \).Practice these steps to enhance your problem-solving skills and tackle similar challenges.
Equation Manipulation
Equation manipulation involves strategically altering equations to isolate variables and solve for unknowns. Algebraic equations provide a method to represent real-world problems mathematically, as seen with \( 5x - 2 = 33 \).To manipulate this equation:
  • Additions and subtractions: Apply inverse operations to both sides to simplify the equation.
  • For instance, we add 2 to both sides to counteract the subtraction.

  • Multiplications and divisions: Similarly, use inverse operations like division to isolate the variable\( x \).
  • In our problem, dividing the entire equation by 5 does the trick.
By altering equations step-by-step, we break down complex expressions into simpler components to find solutions.
Mathematical Operations
Mathematical operations such as addition, subtraction, multiplication, and division are the basic tools used in algebra.They allow us to rearrange and solve equations effectively. Let's take a closer look at how they apply to our problem:
  • Multiplication: The initial step involves multiplying the chosen number by 5. This forms the term \( 5x \) in our equation.

  • Subtraction: Next, we subtract 2 from the product. This subtraction is what Justine instructed, thus adjusting the equation to \( 5x - 2 \).

  • Addition: To solve for \( x \), our first operation reverses the subtraction by adding 2 to both sides, balancing the equation.

  • Division: Finally, divide by 5 to isolate \( x \). By dividing both sides by the same number, we maintain equality and find the initial number.
Understanding these operations helps in both constructing and deconstructing algebraic equations efficiently.

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