Problem 93 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The product of 7.6 and a number is 17 .

Step-by-Step Solution

Answer

The equation is \( 7.6x = 17 \) and \( x \approx 2.24 \).

Step 1: Identify the Given Information

The problem states 'The product of 7.6 and a number is 17'. Here, the number is unknown, so we can let the unknown number be represented by the variable \( x \).

Step 2: Translate Word Problem into an Equation

The word 'product' refers to multiplication, and 'is' refers to equality. Therefore, the sentence can be translated into the equation: \( 7.6 \times x = 17 \).

Step 3: Solve the Equation for the Unknown

To find the value of \( x \), we need to isolate it. This can be done by dividing both sides of the equation by 7.6. Thus, \( x = \frac{17}{7.6} \).

Step 4: Calculate the Value of the Unknown

Perform the division to find \( x \): \( x = \frac{17}{7.6} \approx 2.2368421 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental building blocks in algebra that use numbers, variables, and mathematical operations to represent real-world situations. In the context of our exercise, an algebraic expression is formed by translating the given sentence "The product of 7.6 and a number is 17" into a more mathematical language. Here, the unknown number is represented by a variable, commonly denoted by "\(x\)." This transformation results in the equation: \(7.6 \times x = 17\).

• Variables: Variables are symbols (like \(x\)) used to represent unknown values in expressions and equations. They allow us to generalize problems and find solutions for different scenarios.
• Constants: Numbers like 7.6 and 17 are constants. They remain unchanged throughout the problem.
• Multiplicative Expressions: When we talk about the "product" of numbers and variables, we are referring to a multiplication operation. Here, 7.6 is a constant being multiplied by the variable \(x\).

By understanding algebraic expressions, students can transform real-world verbal descriptions into a format that is easier to manipulate mathematically, ultimately simplifying the problem-solving process.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division are essential tools that help us manipulate numbers and variables in equations. In our problem, the word "product" signifies the multiplication operation.

• Multiplication: This is the operation used in our equation \(7.6 \times x = 17\). It combines the number 7.6 with the variable \(x\) to express a relationship between two values.
• Division: To solve for \(x\), we use the division operation. By dividing both sides of the equation by 7.6, we isolate \(x\), which translates to: \(x = \frac{17}{7.6}\).

Each operation is like a tool in a toolkit. Understanding when and how to use these operations allows you to solve equations and determine the values of unknowns, bridging the gap from problem to solution.

Problem Solving

Problem solving in mathematics involves a series of logical steps to find solutions for various types of equations and inequalities. The current exercise demonstrates a straightforward approach to solving equations using known methods.

• Identifying Key Information: Start by clearly understanding what the problem is stating. In this case, we identified that "the product of 7.6 and a number is 17" meant using multiplication and equality in our equation.
• Translating Words into Equations: Transform verbal descriptions into mathematical equations. This involves recognizing words like "product" and "is" as signals for mathematical operations.
• Solving the Equation: Use appropriate operations to isolate the variable. Here, division helped us find \(x = \frac{17}{7.6}\). Performing this calculation yielded \(x \approx 2.2368421\).

By breaking down a problem into clear steps and using algebraic techniques, you can systematically arrive at a solution. This method is not only applicable to this specific problem but can be extended to a wide variety of mathematical challenges.