Problem 66 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. The difference of a number and two, divided by five

Step-by-Step Solution

Answer

The expression is \(\frac{x - 2}{5}\).

Step 1: Identify the Phrase

The phrase given is "The difference of a number and two, divided by five." We need to construct an algebraic expression based on this description.

Step 2: Write the Expression for the Difference

First, identify "the difference of a number and two." If we let the number be represented by \(x\), then this can be expressed as \(x - 2\).

Step 3: Divide by Five

Now, according to the phrase, this difference \(x - 2\) is then divided by five. This gives us the expression \(\frac{x - 2}{5}\).

Step 4: Simplify if Possible

Observe the expression \(\frac{x - 2}{5}\). Since there are no like terms to combine and no operations to simplify further, this is the simplest form of the algebraic expression.

Key Concepts

Simplifying Expressions

When working with algebraic expressions, simplifying means making the expression easier to understand or work with by reducing it to its simplest form. Here, we were initially given the phrase "the difference of a number and two, divided by five." The first step involved recognizing how to interpret this phrase into an algebraic form.
To simplify an algebraic expression:

• Follow the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).
• Combine like terms - these are terms that have the same variables raised to the same power.
• Perform any possible arithmetic operations.

In our case, with \(\frac{x - 2}{5}\), there were no like terms, and hence, no further simplification was required. Simplification is complete when no further reductions can be made.

Division in Algebra

Division in algebra often involves dividing expressions or variables. In this exercise, we handled "the difference of a number and two" and then divided the result by five, yielding \(\frac{x - 2}{5}\).
Important points to remember about division in algebra:

• Division distributes over addition: \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\).
• Expressions should ideally be in simplified form before performing division.
• When dividing by a term, each part of the numerator should be divisible by that term if possible.

In this particular problem, division by a constant (five) was straightforward because it was applied to the whole expression \(x - 2\) as a unit. Division by variables can often require more intricate steps depending on whether the variables can be simplified or canceled.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions. These are fundamental skills needed to manipulate and solve expressions and equations. In this scenario, writing out the phrase involved several algebraic operations where understanding each step helped in creating the correct expression.
How to manage algebraic operations:

• Addition and subtraction are used typically to combine like terms.
• Multiplication and division require attention to distributing over other operations.
• Expressions should be clear and should follow the order of operations to ensure accuracy.

Here, subtraction was used first to find the difference, followed by division to match the phrase's description. Executing operations in the correct order ensures the mathematical expression reflects the original phrase accurately.