Problem 24 - Prealgebra

Question

If you begin with an even integer and count by two, you are counting consecutive even integers. Write and solve an equation to find two consecutive even integers whose sum is 50 .

Step-by-Step Solution

Answer

The two consecutive even integers are 24 and 26.

Step 1: Define the Variables

Let the first even integer be represented by \( x \). Since the integers are consecutive even integers, the next integer can be represented as \( x + 2 \).

Step 2: Set Up the Equation

According to the problem, the sum of these two consecutive even integers is 50. Hence, the equation we set up is: \( x + (x + 2) = 50 \).

Step 3: Simplify the Equation

Combine like terms in the equation: \( x + x + 2 = 50 \). This simplifies to \( 2x + 2 = 50 \).

Step 4: Solve for x

To find \( x \), first subtract 2 from both sides of the equation: \( 2x = 48 \). Then, divide both sides by 2: \( x = 24 \).

Step 5: Identify the Consecutive Even Integers

The first integer is \( x = 24 \). The consecutive even integer is \( x + 2 = 24 + 2 = 26 \). Thus, the two consecutive even integers are 24 and 26.

Key Concepts

Understanding Even Integers

Even integers are numbers that can be divided by 2 without leaving a remainder. These numbers have a specific pattern, where each even integer is 2 units away from the next one.

• Examples include 2, 4, 6, 8, and so on.
• Every even integer can be expressed as 2 times an integer, such as 2 times 1 equals 2 or 2 times 3 equals 6.
• This even spacing makes them predictable when solving problems.

When dealing with consecutive even integers, starting from any even number, the next consecutive even integer is obtained by simply adding 2 to the current integer. For example, if you start at 10, the next is 12.

The Art of Equation Solving

Equation solving is like unraveling a mystery. You have a puzzle - the equation - and your job is to solve for unknown values. In our problem, we have the equation:
\[ x + (x + 2) = 50 \]
Notice:

• The equation starts by adding two expressions: \(x\) and \(x + 2\).
• It states that when summed, these expressions equal 50.

To solve it, we perform operations to isolate the unknown - in our case, \(x\). By combining like terms, we simplify the equation to \(2x + 2 = 50\).
Finally, we use reverse operations, such as subtraction and division, to solve for \(x\). Here, subtract 2 from 50, resulting in 48, and then divide by 2 to find \(x = 24\).

Exploring the Sum of Integers

The sum of integers is simply the result of adding them together. In consecutive even integers, their sum plays a key role in forming an equation. For our given problem, the sum is 50.

• The problem describes two integers, which can be mathematically represented and summed as \(x + (x + 2)\).
• This sum instructs how to write the equation: it is equal to 50 here.

Understanding how summation works helps identify what is required to form and solve equations accurately. It also provides a clear target, making calculations easier to manage.

Mastering Algebraic Representation

Algebraic representation involves using letters and symbols to illustrate mathematical problems. By representing unknown integers with variables, we simplify and solve these problems.

• In our exercise, the first even integer is represented by \(x\).
• The next consecutive even integer is \(x + 2\).

This notation forms the foundation for algebraic equations. When the problem specifies a sum, translating verbal descriptions into algebraic expressions like \(x + (x + 2) = 50\) turns language into math.
Such representation makes it easier to manipulate equations, apply mathematical rules, and eventually determine unknown values.