Problem 41 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two consecutive integers is \(77 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 38 and 39.

Step 1: Define the consecutive integers

Let the first integer be denoted as x. Since the integers are consecutive, the next integer will be x + 1.

Step 2: Set up the equation

Given that the sum of the two consecutive integers is 77, the equation will be: x + (x + 1) = 77.

Step 3: Simplify the equation

Combine the like terms to simplify the equation: 2x + 1 = 77.

Step 4: Solve for x.

Subtract 1 from both sides: 2x = 76. Then divide by 2: x = 38.

Step 5: Find the consecutive integers

The first integer is 38. The next consecutive integer is 38 + 1 = 39.

Key Concepts

number word problems

Number word problems are a common way to apply mathematical concepts to real-world scenarios. These problems involve understanding and translating words into mathematical statements. In our exercise, we are given the sum of two consecutive integers and need to determine the integers.

Here is the process for solving it:

• Step 1: Define the integers. If the integers are consecutive, you can use variables to express them. In this exercise, we set the first integer as x and the next as x + 1.
• Step 2: Create an equation. Using the information that their sum is 77, the equation becomes x + (x + 1) = 77.
• Step 3, 4 & 5: Simplify and solve the equation to find the value of x, and then determine the next integer.

This approach helps in transforming words into an algebraic expression, making it easier to solve the problem systematically.

solving linear equations

Let \(x\) be the first integer and then their sum is 77, then the consecutive numbers. Further mathematical skills are required to find these numbers and they are utilized effectively in this linear equations problem.}

basic algebra

Basic algebra involves using variables to represent unknown numbers or quantities in expressions and equations. This problem shows how algebra can simplify the process of finding unknown values.

Here's the breakdown:

• When we say 'let the first integer be x', we introduce a variable. This represents an unknown number that we can solve for.
• The expression x + (x + 1) uses this variable to represent the sum of consecutive integers. Algebra helps us translate words into mathematical equations.
• After setting up the equation x + (x + 1) = 77, we use basic algebraic operations such as combining like terms, subtraction, and division to simplify and solve for x.
• Finally, substituting the value of x back into the expressions helps us find the specific integers involved.

Using these fundamental principles of algebra, we can solve many different kinds of problems systematically. Keep practicing and these techniques will become second nature.