Problem 31 - Elementary Algebra

Question

Set up an algebraic equation and then solve. Twice the sum of two consecutive odd integers is 32 . Find the integers.

Step-by-Step Solution

Answer

The integers are 7 and 9.

Step 1: Understand Consecutive Odd Integers

Let's denote the first consecutive odd integer as \( x \). The next consecutive odd integer will be \( x + 2 \) because odd integers differ by 2.

Step 2: Set Up the Sum of Integers

The sum of the two consecutive odd integers is given by: \[x + (x + 2) = 2x + 2\]

Step 3: Translate the Problem into an Equation

According to the problem, twice the sum of these integers equals 32. We can write this equation as: \[2(2x + 2) = 32\]

Step 4: Solve the Equation

First, distribute the 2 in the equation: \[4x + 4 = 32\] Next, subtract 4 from both sides to isolate the term with \( x \): \[4x = 28\] Finally, divide by 4 to solve for \( x \): \[x = 7\]

Step 5: Find the Consecutive Odd Integers

Since \( x = 7 \), the integers are 7 and 9, where 7 is the first integer and the second integer \( x + 2 = 9 \).

Key Concepts

Consecutive Odd Integers

Odd integers are numbers like 1, 3, 5, and so on. When we speak about consecutive odd integers, we mean odd integers that follow one another directly. For example, 3 and 5 are consecutive odd integers. When you're tackling a problem involving consecutive odd integers, it's essential to understand that they are equally spaced by 2.

To represent this in an algebraic equation, we often use the variable representation method. Let's say you start with the first odd integer as \( x \). The next consecutive odd integer would not be \( x + 1 \), but rather \( x + 2 \), because you've moved two steps forward to maintain their odd property. Continuing from there, a third consecutive odd integer would be \( x + 4 \), and so forth. Developing this understanding is crucial for setting up your equations correctly.

Solving Equations

Once you've formed an equation using variables, solving it involves a few systematic steps that will lead you to find the values of the unknowns. For this specific problem, you've already set up the equation as \(2(2x + 2) = 32\) by recognizing that twice the sum of two consecutive odd integers equals 32.

First, start by expanding the equation to remove the parentheses:

• Distribute the 2 across the terms in the parentheses to result in \(4x + 4 = 32\).

This gives you a linear equation which you can solve using basic algebraic techniques:

• Subtract 4 from both sides to start isolating the variable: \(4x = 28\).
• Divide both sides by 4 to solve for \( x \): \(x = 7\).

Solving equations often involves performing inverses or "undoing" operations. Subtracting balances additions, while dividing counteracts multiplication. These steps methodically lead to solving for the unknown variable.

Translating Word Problems into Equations

Translating word problems into algebraic equations is a skill that requires practice but is incredibly useful in math and daily life alike. The goal is to convert words into mathematical symbols and relations that you can manipulate algebraically.

Let's break down the process:

• Identify keywords and their numerical implications. For instance, "twice" implies multiplying by 2, while "sum" indicates addition.
• Translate known values and relationships into mathematical expressions.
• Set up your equation based on these expressions. In our problem, the statement "twice the sum of two consecutive odd integers is 32" becomes \(2(x + (x + 2)) = 32\).

Crafting an equation from a word problem involves understanding both the mathematical and linguistic context. It requires you to engage with both sides of your brain to ensure you comprehend what's being asked and then represent it accurately in numerical form. This skill is invaluable and applicable to various real-world scenarios.