Problem 49 - Intermediate Algebra

Question

In the following exercises, translate the phrases into algebraic expressions. (a) five times the sum of \(3 x\) and \(y\) (b) the sum of five times \(3 x\) and \(y\)

Step-by-Step Solution

Answer

(a) \[ 5(3x + y) \] (b) \[ 15x + y \]

Step 1: Understand the Phrase for Part (a)

The phrase 'five times the sum of 3x and y' means to first add 3x and y and then multiply the result by 5.

Step 2: Construct the Algebraic Expression for Part (a)

Write the sum of 3x and y inside parentheses: \[ (3x + y) \] Then, multiply the entire sum by 5: \[ 5(3x + y) \]

Step 3: Understand the Phrase for Part (b)

The phrase 'the sum of five times 3x and y' means to multiply 3x by 5 first and then add y to the result.

Step 4: Construct the Algebraic Expression for Part (b)

First, multiply 3x by 5: \[ 5 \times 3x = 15x \] Then, add y to the result: \[ 15x + y \]

Key Concepts

algebraic translation

Translating algebraic expressions is like decoding a secret message. It involves converting words or phrases into algebraic terms. Understanding the order of operations is crucial here.

For example, the phrase 'five times the sum of 3x and y' in part (a) means we should add 3x and y first and then multiply by 5. This tells us we need parentheses to show that the addition happens before the multiplication. The expression becomes: \(5(3x + y)\).

In part (b), the phrase 'the sum of five times 3x and y' implies a different order. Here, we multiply 3x by 5 first which gives us 15x, and then we add y. This gives us the expression: \(15x + y\).

Breaking down these phrases into steps is key to accurate algebraic translation.

mathematical phrases

Mathematical phrases often tell you what operations to perform and in what order. Recognizing keywords in these phrases can help translate them into algebraic expressions.

Keywords like 'sum' indicate addition, while 'product' points to multiplication. Words such as 'times' and 'of' can also signify multiplication.

For instance, 'sum of five times 3x and y' includes 'sum' (addition) and 'five times' (multiplication).

It's equivalent to translating a sentence from one language to another. Once you are familiar with these keywords and their meanings, converting phrases to algebra becomes straightforward.

algebraic expressions

An algebraic expression represents a number or a quantity in terms of variables and constants. These expressions are constructed using various operations like addition, subtraction, multiplication, and division.

For example, in the expression \(5(3x + y)\), 5 is a constant, and 3x and y are variable terms. The operation involved here is multiplication outside the parentheses and addition inside them.

Algebraic expressions can have one or multiple terms. In \(15x + y\), '15x' and 'y' are two separate terms combined by addition. Understanding how to construct these expressions can simplify complex problems and help solve real-world issues.

Remember, quality of construction in algebraic expressions lies in accuracy and the correct order of operations.