Problem 48 - Elementary Algebra

Question

For the following problems, rewrite each phrase using algebraic notation. \(x\) divided by \((7\) times \(b)\)

Step-by-Step Solution

Answer

Question: Rewrite the given phrase using algebraic notation: "x divided by (7 times b)" Answer: The algebraic expression for the given phrase is $$\frac{x}{7b}$$.

Step 1: Identify the variables and constants

In the given phrase, "\(x\) divided by \((7\) times \(b)\)", we have 3 elements: 1. The variable \(x\) 2. The constant 7 3. The variable \(b\)

Step 2: Identify the operations

The given phrase indicates two operations: 1. Division: "\(x\) divided by" 2. Multiplication: "\(7\) times \(b\)"

Step 3: Apply the operations

First, we need to do the multiplication. We multiply 7 by \(b\). In algebra, multiplication between a constant and a variable is usually written without any symbol or with a dot in between. We write this multiplication as \(7b\). The second operation is division. Now we need to divide \(x\) by the result of the multiplication. Division is represented by the fraction line, which is a horizontal line between the numerator (the number being divided) and the denominator (the number you are dividing by).

Step 4: Write the final expression

Now that we have applied the operations, the final algebraic expression for the given phrase is: $$ \frac{x}{7b} $$ This represents "\(x\) divided by \((7\) times \(b)\)" in algebraic notation.

Key Concepts

Variables

In the context of algebra, variables are symbols that represent unknown values or quantities. They act as placeholders that can be substituted with numbers from the real world. In our exercise, we encounter two variables: \(x\) and \(b\).
\(x\) is typical in algebraic expressions and can stand for any number. Similarly, \(b\) is another variable that can represent different values in various contexts.
When you see an expression like \(x\) or \(7b\), remember these are not fixed numbers but can change depending on the situation.

Division

In mathematics, division is the process of distributing a number into equal parts. In our expression, "\(x\) divided by \((7 \times b)\)", division is performed using the fraction format.
The fraction line (or division bar) separates the numerator (\(x\)) from the denominator (\(7b\)). This means we are dividing the value represented by \(x\) by the product of \(7\) and \(b\).
Remember, the placement of the terms matters. The numerator goes above the line and the denominator goes below, clearly showing which number divides the other.

Multiplication

In algebra, multiplication signifies combining numbers or variables together. We often write it next to the terms involved without an explicit symbol.
In our expression "\(7\) times \(b\)", we perform multiplication between the constant \(7\) and the variable \(b\).
This results in the product represented by \(7b\). It's a compact way to show that each unit of \(b\) is multiplied by seven.

Constants

In algebraic expressions, constants are numbers that have a fixed value. They are different from variables, which can change.
For our exercise, the constant is \(7\). This means no matter what values \(x\) and \(b\) are, \(7\) remains the same.
Constants like \(7\) determine the scale of multiplication in expressions like \(7b\), showing how \(b\) is being scaled or multiplied by a fixed number.