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Chapter 7: Problem 20

If 4 is added to twice a number and this sum is multiplied by \(2,\) the result is the same as if the number is multiplied by 3 and 4 is added to the product. What is the number?

Short Answer

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

Step by step solution

01

Define the Variable

Let the unknown number be denoted by the variable .
02

Set Up the First Expression

According to the problem, twice the number added to 4 and then multiplied by 2 can be written as: \(2 \times (2n + 4)\).
03

Set Up the Second Expression

The problem also states that the number multiplied by 3 added to 4 gives the same result, which can be written as: \(3n + 4\).
04

Set Up the Equation

Set the two expressions equal to each other:\[2 \times (2n + 4) = 3n + 4\].
05

Distribute and Simplify

Distribute the 2 on the left side: \[4n + 8 = 3n + 4\].
06

Isolate the Variable

Subtract 3n from both sides to get \[n + 8 = 4\]. Then subtract 8 from both sides to find \[n = -4\].
07

Verify the Solution

Substitute \(-4\) back into the original expressions to check: \[2(2(-4) + 4) = 2(-8 + 4) = 2(-4) = -8\] and \[3(-4) + 4 = -12 + 4 = -8\]. Both sides equal -8, so the solution is correct.

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Key Concepts

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

Defining Variables
When faced with an algebra problem, the first step is to identify what you're solving for. This unknown element is called the variable. In our example, we need to find an unknown number. To keep things simple, we use a letter to represent this unknown number. Here, we'll use the letter 'n' as our variable. This step lays the foundation for setting up the rest of the equation, as 'n' will appear throughout our calculations.
Setting Up Equations
Once the variable is defined, our next step is to translate the problem into mathematical expressions. The problem states: If 4 is added to twice the number and this sum is multiplied by 2, the result is the same as if the number is multiplied by 3 and 4 is added to the product. We break this down into two expressions:
  • Twice the number added to 4 and then multiplied by 2:
    • \[ 2 \times (2n + 4) \]
  • The number multiplied by 3 added to 4:
    • \[ 3n + 4 \]

Now, set up the equation by equating these two expressions, as specified by the problem: \[ 2 \times (2n + 4) = 3n + 4 \]
Isolating Variables
With the equation set up, the next step involves isolating the variable 'n.' This means we need to get 'n' by itself on one side of the equation. Start by distributing the terms on the left side: \[ 4n + 8 = 3n + 4 \] Next, eliminate 3n from both sides:
\[ 4n + 8 - 3n = 3n + 4 - 3n \] which simplifies to
\[ n + 8 = 4 \] To isolate 'n,' subtract 8 from both sides: \[ n + 8 - 8 = 4 - 8 \]
This gives us \[ n = -4 \]
Distributing Terms
Distribution is a key algebraic principle where we multiply a term outside the parenthesis with each term inside the parenthesis. It helps in breaking down complex expressions. In the given problem, after defining our variables and setting up our equation, we have: \[ 2 \times (2n + 4) = 3n + 4 \] Here, we distribute the 2 across the terms inside the parentheses to break it down:
\[ 2 \times 2n + 2 \times 4 \] which simplifies to \[ 4n + 8 \]
Now our equation looks like this:
\[ 4n + 8 = 3n + 4 \] This makes it easier to isolate the variable 'n' and solve the equation.

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Most popular questions from this chapter

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x=y-4\) \(4 x+4=2 y\)

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