Problem 78 - Elementary Algebra

Question

The sum of three consecutive even integers is 72 . Find the integers.

Step-by-Step Solution

Answer

The integers are 22, 24, and 26.

Step 1: Define the Variables

Let's define the first even integer as \( x \). The next two consecutive even integers can be represented as \( x+2 \) and \( x+4 \).

Step 2: Set Up the Equation

Based on the problem, the sum of these three integers is 72. Therefore, we can write the equation: \[ x + (x + 2) + (x + 4) = 72 \].

Step 3: Combine Like Terms

Combine the terms in the equation: \( x + x + x + 2 + 4 = 72 \). This simplifies to \( 3x + 6 = 72 \).

Step 4: Solve for x

Subtract 6 from both sides of the equation to isolate the terms with \( x \): \( 3x = 66 \). Next, divide both sides by 3 to solve for \( x \): \( x = 22 \).

Step 5: Find the Integers

Since \( x = 22 \), the three consecutive even integers are \( x = 22 \), \( x+2 = 24 \), and \( x+4 = 26 \).

Step 6: Verify the Solution

Add the integers to check if they sum to 72: \( 22 + 24 + 26 = 72 \). The solution is verified.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in a sequence where each number differs from the next by a constant value. In this exercise, we're dealing with consecutive even integers. These numbers differ by 2 because every even number is separated by 2 from the next. For example, if you start with the number 2, the next consecutive even numbers would be 4, 6, 8, and so on.

• In the problem, we defined three consecutive ones: if the first is \( x \), the others would be \( x+2 \) and \( x+4 \).
• This sequence aids in setting equations because it provides a straightforward way to express each number relative to the first.

Understanding this is crucial in algebra as it sets the foundation for creating and solving equations involving sequences.

Equations

Equations in algebra are mathematical statements that assert the equality of two expressions. In algebra problems, setting up an equation is a step towards finding the solution. This process usually follows from the definitions of the variables and the problem conditions.

• Here, the sum of the three consecutive even integers is set up as an equation: \( x + (x+2) + (x+4) = 72 \).
• The equation serves as a mathematical representation of the problem and it is vital for further problem solving.

Once you've set up your equation, you can perform operations to simplify it and solve for the unknowns.

Variables

In algebra, variables are symbols used to represent unknown values. They provide a way to describe general relationships and solve problems involving numbers.

• In this task, we use \( x \) for the first even integer as our variable.
• This helps us express the subsequent even integers as \( x+2 \) and \( x+4 \), simplifying the problem.

Using variables allows us to create equations and systematically solve them, highlighting our ability to handle unknowns through algebraic methods.

Problem Solving

Problem solving in algebra often involves breaking down a problem into familiar parts, using pre-defined steps.

• Firstly, identify what needs to be found, which in this case was the three consecutive even integers.
• Next, define your variables and set up your equation based on problem descriptions.
• Simplify and manipulate the equation to solve for the unknown variable.
• Finally, substitute back to verify your solutions are correct and make sense.

Effective problem solving is essential for any algebra exercise, as it encourages systematic thinking and aids in finding solutions accurately and efficiently.