Problem 77 - Elementary Algebra

Question

The sum of three consecutive odd integers is \(45 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 13, 15, and 17.

Step 1: Define the unknowns

Let the first odd integer be \( x \). Since the integers are consecutive and odd, the second integer can be expressed as \( x + 2 \) and the third as \( x + 4 \).

Step 2: Set up the equation

The problem states that the sum of these three integers is 45. So we can set up the equation: \( x + (x + 2) + (x + 4) = 45 \).

Step 3: Simplify the equation

Combine the like terms in the equation: \( 3x + 6 = 45 \).

Step 4: Solve for \( x \)

Subtract 6 from both sides of the equation to isolate the term with \( x \): \( 3x = 39 \). Then divide both sides by 3 to solve for \( x \): \( x = 13 \).

Step 5: Find the consecutive integers

Now that we have \( x = 13 \), substitute back to find the integers: the first integer is 13, the second is \( 13 + 2 = 15 \), and the third is \( 13 + 4 = 17 \).

Step 6: Verify the solution

Check that the sum of 13, 15 and 17 is indeed 45: \( 13 + 15 + 17 = 45 \). Since this is true, the solution is verified.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the value of unknown variables. In the given problem, we used an equation to find three consecutive odd integers whose sum is 45.

Solving an equation usually involves a few key steps:

• Identifying the unknown(s) by assigning a variable, such as \( x \).
• Setting up an equation based on the given problem, such as \( x + (x + 2) + (x + 4) = 45 \) for our example.
• Simplifying the equation by combining like terms, leading to \( 3x + 6 = 45 \).
• Isolating the variable on one side to solve for it, as we did by first subtracting 6 from both sides resulting in \( 3x = 39 \), and then dividing by 3 to find \( x = 13 \).

Solving equations is like peeling an onion. You systematically strip away layers to reveal the core solution, making sure each step is logically sound.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. In context, they are used to represent relationships between quantities. For solving the problem of finding consecutive odd integers, understanding these expressions is key.

We start with defining the integers as algebraic expressions:

• The first odd integer is \( x \).
• The second is represented as \( x + 2 \), capturing that it is the next odd number.
• Similarly, the third is \( x + 4 \).

Each expression builds upon the first by adding a consistent term, which in this case is 2, the difference between consecutive odd numbers.

These expressions are combined in an equation which is then manipulated using algebraic rules. Writing and simplifying these expressions is essential, as it sets a clear path to solving the equations.

Integer Properties

Understanding integer properties, particularly those of odd integers, is crucial for this exercise. Consecutive odd integers are integers that follow each other in sequence, each differing by 2. This distinct pattern ensures one integer is always placed after the odd-numbered preceding it.

Here are some key properties:

• Odd integers like 1, 3, 5, 7, etc., increase by 2 when consecutive.
• They always yield an odd sum operation when added to another odd number.
• In our problem, this relationship helped express the integers as \( x, x+2, \text{and} x+4 \).

By knowing these properties, we can confidently set the expressions for odd numbers and use them in the equation. Integer properties not only define the structure of numbers in math but also guide us in creating and solving equations efficiently.