Problem 21 - Algebra 2

Question

Write an algebraic expression to represent each verbal expression. five times the sum of 9 and a number

Step-by-Step Solution

Answer

The algebraic expression is \(5(9 + x)\).

Step 1: Understand the Verbal Expression

The verbal expression given is "five times the sum of 9 and a number." This needs to be translated into an algebraic expression.

Step 2: Identify the Components

In the expression, we have "the sum of 9 and a number." The number is unknown and can be represented by a variable, commonly 'x.' So, the sum of 9 and a number is represented as \(9 + x\).

Step 3: Translate 'Times' into Multiplication

The word 'times' implies multiplication. The phrase "five times" means that we need to multiply 5 by the sum we identified in the previous step.

Step 4: Write the Algebraic Expression

Put it all together by multiplying 5 with the sum \(9 + x\): \[ 5 imes (9 + x) \] This represents 'five times the sum of 9 and a number.'

Key Concepts

Verbal Expressions

Verbal expressions are phrases that describe mathematical operations in words. They are crucial when translating everyday language into mathematical language. For example, phrases like "five times," "the sum of," or "a number" give us hints about the arithmetic operations involved. Understanding these verbal cues is key to forming accurate algebraic expressions.
In the given exercise, the phrase "five times the sum of 9 and a number" is our verbal expression. Here, "five times" indicates multiplication, "the sum of" indicates we need to add, and "a number" suggests the need for a variable. By identifying these parts, we can begin to convert verbal expressions into algebraic ones.

Translation of Expressions

Learning to translate expressions from words to algebra is like learning a new language. It involves recognizing the arithmetic operations implied by words. One must grasp the relationship between words like 'sum' for addition and 'times' for multiplication.
The translation process involves breaking down the verbal expression into smaller parts. First, interpret the 'sum of 9 and a number.' This translates to the mathematical statement: \( 9 + x \), where \( x \) is an unknown variable. Next, 'five times' guides the multiplication of 5 by this sum. So the complete translation converts into: \[ 5 \times (9 + x) \]
Understanding how to translate such expressions is essential for solving real-world problems using algebra.

Algebra

Algebra is the branch of mathematics where symbols and letters represent numbers and quantities in formulas and equations. It is a powerful tool that allows us to generalize mathematical concepts. Here's why it matters:

• **Variables**: Represent unknown quantities. In our expression, \( x \) is a placeholder for a number we do not yet know.
• **Constants**: Numbers that are always the same, such as 9 in our problem.
• **Operations**: Such as addition and multiplication, which are represented by their corresponding arithmetic symbols.

By using algebra, we can describe relationships and solve equations that help figure out unknown values. Understanding the basics of algebra opens the door to more complex mathematical reasoning and problem solving.

Mathematical Reasoning

Mathematical reasoning is the process of thinking logically and systematically in order to understand and solve problems. It involves the ability to make connections between different concepts and use them effectively.
Applying mathematical reasoning when dealing with expressions involves:

• **Understanding the problem**: Recognize what is being asked or described in a problem.
• **Devising a plan**: Identify what operations or calculations are necessary.
• **Carrying out the plan**: Execute the calculations or manipulations correctly.
• **Reviewing the solution**: Check that the answer makes sense within the context of the problem.

In our exercise, we demonstrated mathematical reasoning by translating a verbal problem ("five times the sum of 9 and a number") into an algebraic expression \( 5 \times (9 + x) \), using logic to ensure each step aligns correctly with the intended operations. This skill is vital for not only math tasks but also in real-world problem solving.