Problem 58 - Intermediate Algebra

Question

Explain the difference between the phrases "4 times the sum of \(x\) and \(y^{\prime \prime}\) and "the sum of 4 times \(x\) and \(y^{\prime \prime}\).

Step-by-Step Solution

Answer

The first phrase means multiplying the sum of \(x\) and \(y^{\backprime \backprime}\) by 4, while the second phrase means adding \(y^{\backprime \backprime}\) to 4 times \(x\).

Step 1: Understanding the First Phrase

The first phrase is '4 times the sum of \(x\) and \(y^{\backprime \backprime}\)'. This means you need to first find the sum of \(x\) and \(y^{\backprime \backprime}\), then multiply that result by 4. Mathematically, this can be expressed as \[4 \times (x + y^{\backprime \backprime})\].

Step 2: Understanding the Second Phrase

The second phrase is 'the sum of 4 times \(x\) and \(y^{\backprime \backprime}\)'. This means you first need to calculate 4 times \(x\), and then add \(y^{\backprime \backprime}\) to that product. Mathematically, this can be expressed as \[4x + y^{\backprime \backprime}\].

Step 3: Comparing Both Expressions

Compare the two mathematical expressions: \[4 \times (x + y^{\backprime \backprime})\] and \[4x + y^{\backprime \backprime}\]. Notice that \(4 \times (x + y^{\backprime \backprime})\) indicates that the sum of \(x\) and \(y^{\backprime \backprime}\) is multiplied by 4, while \(4x + y^{\backprime \backprime}\) indicates that only \(x\) is multiplied by 4 and then \(y^{\backprime \backprime}\) is added to the result.

Key Concepts

Order of Operations

The order of operations is key when interpreting mathematical expressions. It dictates the sequence in which different operations should be carried out. The order is commonly remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

Let’s apply this to our phrases. For '4 times the sum of \(x\) and \(y''\)', you first perform the operation inside the parentheses (the sum of \(x + y''\)) and then multiply the result by 4, giving us \[4 \times (x + y'')\]. For 'the sum of 4 times \(x\) and \(y''\)', you separately calculate 4 times \(x\) (which is \[4x\]), and then add \(y''\) to it: \[4x + y''\].
Understanding the order of operations helps to clearly differentiate these phrases, ensuring correct implementation of arithmetic rules.

Algebraic Interpretation

Algebraic interpretation involves translating word phrases into algebraic expressions. This is crucial for solving problems correctly. In our example, the first phrase '4 times the sum of \(x\) and \(y''\)' requires us to encapsulate the sum \(x + y''\) inside parentheses because the sum is considered a single entity before multiplication. Its algebraic representation is \[4 \times (x + y'')\].
The second phrase 'the sum of 4 times \(x\) and \(y''\)' divides the operations into two separate parts: multiplying \(x\) by 4 (yielding \[4x\]) and then adding \(y''\) to this result: \[4x + y''\].
This clear demarcation of operations aids in accurately expressing and solving algebraic problems.

Mathematical Expressions

Mathematical expressions are ways to clearly present mathematical ideas and operations through symbols and numbers. Examining our given phrases, we convert them into precise expressions. '4 times the sum of \(x\) and \(y''\)' turns into \[4 \times (x + y'')\], clarifying that the sum happens first, followed by the multiplication.
For the phrase 'the sum of 4 times \(x\) and \(y''\)', we break it down to first calculate \[4x\] and then add \(y''\). The resulting expression, \[4x + y''\], identifies each separate mathematical operation.
A good grasp of forming and interpreting expressions ensures correct problem-solving in algebra.