Problem 44 - Intermediate Algebra

Question

In the following exercises, translate the phrases into algebraic expressions. (@) the difference of \(17 x^{2}\) and \(5 x y\) (b) the quotient of \(8 y^{3}\) and \(3 x\) (C) Eighteen more than \(a^{2}\); (1) \(11 b\) less than \(100 b^{2}\)

Step-by-Step Solution

Answer

\(17x^{2} - 5xy\), \(\frac{8y^{3}}{3x}\), \(a^{2} + 18\), \(100b^{2} - 11b\)

Step 1: Identify the components

Recognize the parts of each phrase that need to be translated into algebraic expressions. For each part, determine what mathematical operation is being described.

Step 2: Translate the difference of 17x² and 5xy

The phrase 'the difference of' indicates a subtraction operation. Thus, translate 'the difference of 17x² and 5xy' to the expression: \(17x^{2} - 5xy\)

Step 3: Translate the quotient of 8y³ and 3x

The phrase 'the quotient of' indicates a division operation. Thus, translate 'the quotient of 8y³ and 3x' to the expression: \(\frac{8y^{3}}{3x}\)

Step 4: Translate Eighteen more than a²

The phrase 'more than' indicates an addition operation. Thus, translate 'Eighteen more than a²' to the expression: \(a^{2} + 18\)

Step 5: Translate 11b less than 100b²

The phrase 'less than' indicates a subtraction operation. Thus, translate '11b less than 100b²' to the expression: \(100b^{2} - 11b\)

Key Concepts

Difference in Algebra

In algebra, the word 'difference' typically means we are dealing with a subtraction operation. To effectively translate verbal phrases into algebraic expressions, it's important to understand that 'difference' signifies the operation of subtracting one term from another.
For instance, in the exercise 'the difference of 17x² and 5xy', we need to subtract 5xy from 17x². The correct algebraic expression becomes: \[ 17x^{2} - 5xy \]. To further clarify, here are a few tips:

• Always identify the terms involved.
• Make sure you understand which term is being subtracted from which.
• Write the expression clearly to avoid confusion.

Understanding these steps will help you master the concept of 'difference' in algebra.

Quotient in Algebra

In algebra, the 'quotient' refers to the result of a division operation. When translating phrases that include the word 'quotient', you should look for the terms to be divided and then represent them in fraction form.
For example, the exercise 'the quotient of 8y³ and 3x' translates to:\[ \frac{8y^{3}}{3x} \]. Here are some useful tips to keep in mind:

• Identify the numerator (the term being divided).
• Identify the denominator (the term by which to divide).
• Ensure you write the division as a fraction to clearly show the quotient.

Thoroughly understanding these elements helps you accurately translate and work with 'quotient' in algebraic expressions.

Addition in Algebra

The term 'addition' in algebra denotes the sum of two or more terms. Phrases like 'more than', 'plus', or 'increased by' indicate that you need to perform an addition operation.
Let's examine the exercise 'Eighteen more than a²'. This means we need to add 18 to a², giving us the expression: \[ a^{2} + 18 \]. Here are a few quick tips:

• Identify the base term (in this case, a²).
• Determine the term to be added (here, it's 18).
• Write the expression clearly to show the addition.

With these guidelines, you'll find it easier to work with 'addition' in algebra.

Subtraction in Algebra

Subtraction in algebra is signified by terms like 'less than', 'minus', or 'subtract'. It involves deducting one term from another. Understanding which term comes first and which is being subtracted is crucial.
In the exercise '11b less than 100b²', the phrase 'less than' means we need to subtract 11b from 100b², resulting in:\[ 100b^{2} - 11b \]. Here are some tips to make it easier:

• Identify the minuend (the number from which you are subtracting, here it's 100b²).
• Identify the subtrahend (the number to be subtracted, here it's 11b).
• Write the expression clearly to reflect the subtraction.

Following these tips will help you correctly handle 'subtraction' in algebra.