Problem 24 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The sum of three consecutive even integers is 96 . Find the integers.

Step-by-Step Solution

Answer

The integers are 30, 32, and 34.

Step 1: Understanding Consecutive Even Integers

First, let's introduce variables for the three consecutive even integers. If the first integer is \( x \), then the next two consecutive even integers can be expressed as \( x+2 \) and \( x+4 \).

Step 2: Setting Up the Equation

According to the problem, the sum of these three consecutive even integers is 96. We can set up the equation based on this information: \( x + (x + 2) + (x + 4) = 96 \).

Step 3: Simplifying the Equation

Now, combine like terms in the equation: \( 3x + 6 = 96 \).

Step 4: Solving for x

Subtract 6 from both sides of the equation to isolate the term with \( x \): \( 3x = 90 \). Then, divide both sides by 3 to find \( x \): \( x = 30 \).

Step 5: Identifying the Integers

With \( x = 30 \), the integers are 30 (\( x \)), 32 (\( x+2 \)), and 34 (\( x+4 \)).

Key Concepts

Algebraic Equation

An algebraic equation is a mathematical statement that uses variables, numbers, and operations to express a relationship. It typically contains an equal sign, indicating that two expressions are the same. In our exercise, we set up an equation to represent the sum of three consecutive even integers. This equation takes the form of a linear expression:

• First integer: \( x \)
• Second integer: \( x + 2 \)
• Third integer: \( x + 4 \)

With the sum provided as 96, we translate this into an equation: \( x + (x + 2) + (x + 4) = 96 \). The key here is to represent the situation using a formal algebraic structure, allowing us to solve for the unknown variable.

Consecutive Even Integers

Consecutive even integers are numbers in sequence where each number is 2 units apart from the next one. Unlike general consecutive integers, even integers follow a unique pattern of having a fixed difference of 2. For instance, in the sequence 2, 4, 6, each number continues by adding 2 to the previous number.When expressing this algebraically, if the first integer is \( x \), the following consecutive even integers can be represented as:

• Second integer: \( x + 2 \)
• Third integer: \( x + 4 \)

Understanding this concept helps in setting up proper equations. By assigning a variable to the first integer, the arithmetic pattern translates into an easy-to-solve algebraic form.

Solving Equations

Solving an equation involves finding the value of the variable that makes the equation true. Here, we start with the equation from our problem: \( x + (x + 2) + (x + 4) = 96 \). First, simplify by combining like terms:

• Combine: \( x + x + x = 3x \)
• Combine constants: \( 2 + 4 = 6 \)

The simplified equation becomes \( 3x + 6 = 96 \). To solve for \( x \), we need to isolate it:1. Subtract 6 from both sides: \( 3x = 90 \)2. Divide both sides by 3: \( x = 30 \)These steps effectively find the value of \( x \), completing the primary goal of solving the equation.

Integer Sums

Integer sums relate to the idea of adding whole numbers together. In the context of consecutive even integers, our exercise was to find three numbers whose sum is 96.The given integers are expressed in terms of \( x \): 30, 32, and 34. Mathematics allows us to set up and manipulate expressions to understand and solve problems involving the addition of integers. The process includes:

• Representing each integer: \( x, x+2, x+4 \)
• Formulating their sum: \( x + (x + 2) + (x + 4) = 96 \)

This systematic addition showcases how structured algebraic methods can solve questions about integer sums and verify solutions through calculation.