Problem 55 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. \(s\) subtracted from \(S\) b. \(S\) subtracted from \(s\)

Step-by-Step Solution

Answer

a. \(S - s\) b. \(s - S\)

Step 1: Identify Operation for Part A

The phrase 's subtracted from S' indicates we need to subtract the variable \(s\) from the variable \(S\). This implies we perform a subtraction operation where \(s\) is taken from \(S\).

Step 2: Write Expression for Part A

The correct algebraic expression is \(S - s\). This represents \(s\) being subtracted from \(S\).

Step 3: Identify Operation for Part B

For 'S subtracted from s', we understand it as the variable \(S\) being subtracted from the variable \(s\). This requires a subtraction operation reversing the order of the variables as indicated in Part A.

Step 4: Write Expression for Part B

The algebraic expression reflecting the phrase 'S subtracted from s' is \(s - S\). This signifies \(S\) being subtracted from \(s\), the reverse order from Part A.

Key Concepts

Translation of Phrases into Algebraic Expressions

Translating phrases into algebraic expressions is an essential skill in algebra. It involves identifying the mathematical operations described by words and converting them into mathematical symbols. For example:

• The word "sum" usually implies addition.
• "Product" indicates multiplication.
• "Subtracted from," as seen in our exercise, indicates a subtraction operation where the order is crucial.

To tackle these problems effectively, pay attention to each word and its placement. For instance, when "s subtracted from S" appears, it implies that you start with "S" and remove "s." This gives you the expression \( S - s \). The phrase "from" in subtraction tells us the order of the variables—remember, the first term mentioned is the one from which you subtract.

Subtraction in Algebra

Subtraction in algebra operates similarly to subtraction in arithmetic, but it involves variables. Recognizing the significance of order is vital in algebraic subtraction because it affects the result. Unlike addition, subtraction is not commutative, meaning

• \( a - b \) is not the same as \( b - a \).
• The outcome depends heavily on which term is being subtracted from which.

In our exercise, the phrases dictate the order:

• "s subtracted from S" translates to \( S - s \).
• Conversely, "S subtracted from s" results in \( s - S \).

This order reversal drastically changes the result, which illustrates the importance of carefully interpreting phrases when writing algebraic expressions.

Role of Variables in Algebra

In algebra, variables are symbols, usually letters, that represent numbers whose values are not yet known or can change. They are the backbone of algebraic expressions, allowing us to write general formulas that can work for many possible values. When translating phrases into algebraic expressions, each word is critical in defining how variables like "s" and "S" interact. In our exercise:

• "s" and "S" are variables that stand in for specific values or quantities.
• They form expressions that can be dynamically adjusted depending on their assigned values.

Understanding the variables' roles helps us to grasp the overall picture of an algebraic problem. With this knowledge, it becomes easier to manipulate the expressions and solve equations that involve these variables. Moreover, knowing how to use variables enables you to explore ways to represent real-world situations mathematically.