Problem 81 - Algebra A Combined Function

Question

Write each algebraic expression described. Simplify if possible. See Example \(11 .\) Classrooms on one side of the science building are all numbered with consecutive even integers. If the first room on this side of the building is numbered \(x,\) write an expression in \(x\) for the sum of five classroom numbers in a row. Then simplify this expression.

Step-by-Step Solution

Answer

The simplified expression for the sum of the five consecutive even classroom numbers is \(5x + 20\).

Step 1: Identify the Pattern

The classroom numbers are consecutive even integers starting from \(x\). Even integers increase by 2, so the numbers are \(x, x+2, x+4, x+6, x+8\).

Step 2: Write the Expression for the Sum

To find the sum of the classroom numbers, add them together: \(x + (x+2) + (x+4) + (x+6) + (x+8)\).

Step 3: Simplify the Expression

Combine the like terms in the expression: \(x + x + x + x + x + 2 + 4 + 6 + 8 = 5x + 20\).

Key Concepts

Consecutive Integers

Consecutive integers follow one after the other in equal steps, often by 1 or certain multiples. In our particular case involving even numbers, consecutive integers increase by 2. For example, starting from a number 6: the next consecutive even integers would be 8, 10, 12, etc.
It's quite useful to write them as algebraic expressions to easily handle sums or other operations. When starting from any variable like \( x \), representing consecutive even integers would be \( x, x+2, x+4, x+6, \) and so on.
This manner of expressing numbers is especially helpful as it allows manipulation of these values through algebraic operations like addition, subtraction, and more.

• In practice, this helps simplify tasks like finding sums, by directly incorporating expressions.
• It aids in solving real-world problems where sequences of evenly spaced numbers are involved.

Simplifying Expressions

Expressions in algebra can look complicated but can be simplified to help you understand and solve problems more easily. Simplification basically means combining like terms to condense an expression.
For example, if you have the expression \( x + (x+2) + (x+4) + (x+6) + (x+8) \), you're looking at multiple terms which can be grouped.
When simplifying:

• Group all terms involving \( x \).


• Next, add up all constant numbers (those without \( x \)).

After calculations and combining like terms, the expression becomes \( 5x + 20 \).
This implies adding the equivalent value of \( x \) five times, then adding 20 (result of the successive additions of the constants).
Using such techniques, solving and understanding algebra becomes straightforward!

Even Numbers

Even numbers form a particular category of integers divisible by 2, like 2, 4, 6, 8, and so forth. Recognizing these numbers helps in devising expressions for scenarios that involve similar categories, such as situations with column seating, numbering, etc.
In algebraic expressions, even numbers can easily be modeled using a base like \( x \) and adding consecutive increments of 2. This example: \( x, x+2, x+4, \) illustrates consecutive even numbering.

• These expressions signify sequences that can be utilized in counting or arranging settings, like room numbers.


• They are vital for computational tasks where steps of two or more matter significantly.

Such understanding not only aids in breaking down complex math problems but also provides a handy approach for logical reasoning tasks, designed around patterns you often see in real-life situations.