Problem 47 - Intermediate Algebra

Question

In the following exercises, translate the phrases into algebraic expressions. (a) eight times the difference of \(y\) and nine (b) the difference of eight times \(y\) and 9

Step-by-Step Solution

Answer

For (a): \( 8(y - 9) \), For (b): \( 8y - 9 \)

Step 1 Title - Understand the first phrase

The first phrase is 'eight times the difference of y and nine.' This means you need to find the difference between y and 9 first.

Step 2 Title - Write the difference between y and nine

The difference between y and nine can be written as \( y - 9 \).

Step 3 Title - Multiply by eight

Next, multiply the difference by eight, which leads to \( 8 \times (y - 9) \).

Step 1 Title - Understand the second phrase

The second phrase is 'the difference of eight times y and 9.' This means you need to find eight times y first and then subtract 9.

Step 2 Title - Write eight times y

Eight times y can be written as \( 8y \).

Step 3 Title - Subtract 9

Subtract 9 from eight times y, which results in \( 8y - 9 \).

Key Concepts

Translating phrases into algebraic expressions

Translating phrases into algebraic expressions might sound tricky at first, but it's all about breaking down the sentence into smaller parts and understanding the mathematical operations suggested by the words. When you read a phrase like 'eight times the difference of y and nine', you should start by identifying what each part of the phrase represents.
In this case, 'the difference of y and nine' means you're looking for a subtraction operation, which can be written as \(y - 9\).
Then, you are told to take 'eight times' this difference, which means you need to multiply the previous result by eight: \(8 \times (y - 9)\).
By carefully breaking down and understanding each part of the phrase, you can accurately translate English phrases into algebraic expressions.

Mathematical operations

Mathematical operations are fundamental in algebra and allow us to manipulate and transform expressions to solve problems. The primary operations include addition, subtraction, multiplication, and division.
In the example, we encounter several operations: subtraction in 'the difference of y and nine', represented as \(y - 9\), and multiplication in 'eight times', shown as \(8 \times (y - 9)\).
Understanding these operations is crucial because they tell you how to process the numbers and variables. When instructions say 'subtract 9 from eight times y', you perform the multiplication first, yielding \(8y\), and then carry out the subtraction to get \(8y - 9\).
Each of these operations follows a specific order known as the order of operations (PEMDAS/BODMAS), which ensures consistency and clarity in solving equations.

Intermediate algebra problems

Intermediate algebra problems often involve translating complex phrases into algebraic expressions and understanding the sequence of operations to simplify or solve them.
Let's take the second exercise: 'the difference of eight times y and 9.' First, we identify 'eight times y' as \(8y\).
Next, 'the difference of' instructs us to subtract 9 from \(8y\), resulting in the expression \(8y - 9\).
These types of problems are common in intermediate algebra and help build a foundation for more advanced algebraic concepts.
The key to mastering these problems is to practice breaking down phrases step-by-step and applying the correct mathematical operations. With practice, you'll become more comfortable translating phrases into expressions and solving increasingly complex problems.