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Chapter 1: Problem 8

For each problem, familiarize yourself with the situation. Then translate tomathematical language. You need not actually solve the problem; just carry outthe first two steps of the five-step strategy. You will be asked to completesome of the solutions as Exercises 35–42. The sum of two numbers is \(88 .\) One of the numbers is 6 more than the other.What are the numbers?

### Short Answer

Expert verified

Let the numbers be \(x\) and \(x + 6\); the equation is \(x + (x + 6) = 88\).

## Step by step solution

01

## - Define Variables

Let the two numbers be represented by variables. Define the first number as \(x\). Since the second number is 6 more than the first number, define the second number as \(x + 6\).

02

## - Translate into Equations

According to the problem, the sum of the two numbers is 88. Therefore, create the equation \(x + (x + 6) = 88\).

## Key Concepts

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

###### variable definition

In the context of solving mathematical problems, defining variables is the first crucial step. Variables are symbols that represent unknown values. In our exercise, we need to find two numbers that add up to 88, with one number being 6 more than the other. To simplify our work, we assign a variable to the first unknown number. Let’s use the variable \(x\) to represent the first number.

Now, since the second number needs to be 6 more than the first, we define it as \(x + 6\). By using variables, we turn the word problem into a mathematical one, setting ourselves up for translating it into an equation later.

###### translation to equations

Once variables are defined, the next step is translating the word problem into mathematical equations. In this exercise, the problem states the sum of two numbers is 88. From our variable definitions, we know:

- The first number is \(x\)

- The second number is \(x + 6\)

To find their sum, we add these two expressions together. Therefore, the equation representing their sum is:

\(x + (x + 6) = 88\)

By setting up this equation, we’ve successfully translated the word problem into a mathematical relationship. This equation can now be simplified and solved in the problem-solving strategy.

###### problem-solving strategy

Building a strategy for solving the problem involves a series of logical steps. First, we start with our equation:

\(x + (x + 6) = 88\).

Combine like terms to simplify the equation:

\(2x + 6 = 88\).

Next, isolate the term with the variable by subtracting 6 from both sides:

\(2x = 82\).

Finally, solve for \(x\) by dividing both sides by 2:

\(x = 41\).

With these steps completed, we’ve found the first number, which is 41. To find the second number, recall our variable definition: the second number is \(x + 6\). Therefore,

\(41 + 6 = 47\).

In conclusion, the two numbers are 41 and 47. Each step ensures a clear path from defining variables to solving the equation, making linear equations approachable and easy to understand.

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