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Chapter 7: Problem 16

If 3 is added to a number and this sum is doubled, the result is 2 more than the number. Find the number.

Short Answer

Expert verified
The number is \( -4 \).

Step by step solution

01

- Define the Variable

Let the unknown number be represented by the variable \( x \).
02

- Create the Equation

According to the problem, if 3 is added to the number \( x \), we get \( x + 3 \). Then, this sum is doubled, giving us \( 2(x + 3) \).
03

- Formulate the Relationship

The problem states that the doubled sum is 2 more than the number itself. Therefore, the equation can be written as:\[ 2(x + 3) = x + 2 \]
04

- Simplify the Equation

Expand and simplify the equation to isolate \( x \):\[ 2x + 6 = x + 2 \]
05

- Solve for x

Subtract \( x \) and 6 from both sides to solve for \( x \):\[ 2x + 6 - x - 6 = x + 2 - x - 6 \]This simplifies to:\[ x = -4 \]
06

- Verify the Solution

To verify, substitute \( x = -4 \) back into the original condition: Adding 3 to \( -4 \) gives: \( -4 + 3 = -1 \). Doubling this sum: \( 2(-1) = -2 \). According to the problem, this should equal \( -4 + 2 = -2 \), which is correct. Thus, the number is verified as \( -4 \).

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Key Concepts

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

Variable Definition
In mathematics, a variable is a symbol used to represent an unknown quantity. It serves as a placeholder for the value we are trying to determine. In this exercise, we need to find a number, so we define the variable to make it easier to work with the equation. Let the unknown number be represented by the variable \( x \). This allows us to use algebraic methods to solve for \( x \). Defining the variable is the first step in translating a word problem into a mathematical equation.
Equation Formulation
The next step is to create an equation based on the information given in the problem. According to the exercise, if 3 is added to the unknown number \( x \), we get \( x + 3 \). Then, this sum is doubled, resulting in \( 2(x + 3) \). The problem also says that this doubled sum is 2 more than the number itself. Therefore, we can write this relationship as an equation:
  • \[ 2(x + 3) = x + 2 \]
To formulate the equation, we must correctly interpret the problem's language and translate it into a mathematical expression. This step is crucial for solving the problem successfully.
Simplification
Once the equation is formulated, the next step is simplification. This involves expanding and combining like terms to isolate the variable. Starting from the equation: \( 2(x + 3) = x + 2 \), we first expand it:
  • \[ 2x + 6 = x + 2 \]
To isolate \( x \), we need to perform the appropriate algebraic operations. We subtract \( x \) and 6 from both sides of the equation:
  • \[ 2x + 6 - x - 6 = x + 2 - x - 6 \]
This simplifies to:
  • \[ x = -4 \]
Simplification makes the equation easier to solve, providing a clearer path to finding the value of the unknown variable.
Verification
Verification is an important step to ensure that the solution obtained is correct. After solving for \( x \) and finding it to be \( -4 \), we substitute this value back into the original conditions of the problem to check if it holds true. According to the problem, if we add 3 to \( -4 \), we get \( -1 \). Doubling this sum results in:
  • \[ 2(-1) = -2 \]
The problem states that the doubled sum should be 2 more than the number itself, which is true if we compare it with \( -4 + 2 = -2 \). Since the conditions match, the number is correctly verified as \( -4 \). Verification helps ensure that our solution is accurate and consistent with the given problem.

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