Problem 34 - Algebra 1

Question

Write the verbal sentence as an equation, or an inequality. Seventy divided by the product of seven and a number \(p\) is equal to one.

Step-by-Step Solution

Answer

The equation is \( 70 / (7 \cdot p) = 1 \).

Step 1: Identify elements

First, identify the phrases in the exercise and their mathematical representations. 'Seventy' refers to 70, 'divided by' refers to the division operation (/), 'the product of seven and a number \( p \)' indicates multiplication of 7 and \( p \), and 'is equal to one' indicates = 1.

Step 2: Construct equation using mathematical operations

Next, replace the phrases in the exercise with their corresponding mathematical symbols from step 1. This forms the equation: \( 70 / (7 \cdot p) = 1 \).

Key Concepts

Writing Equations

When it comes to writing equations from verbal sentences, the aim is to translate words into a mathematical format that can be used for calculation or problem-solving. The key is understanding the vocabulary and phrases that signify different mathematical operations and constants. For example, the word 'divided by' in a sentence suggests the division operation. A number mentioned in the sentence is treated as a known value or constant. Here, the number seventy conveys the constant 70.

In educational terms, writing equations is a foundational skill in algebra. The process involves identifying numerical and variable components, as well as understanding the meanings of terms such as 'product of' and 'is equal to,' which correspond to multiplication and the equality symbol, respectively.

Mathematical Operations

Understanding mathematical operations is crucial in translating verbal sentences into equations. Operations such as addition, subtraction, multiplication, and division are typically implied through phrases like 'the sum of,' 'difference between,' 'the product of,' and 'divided by.' When writing equations, you need to replace these verbal cues with their mathematical symbols: plus (+), minus (-), times (*), and divided by (/), respectively.

In the given exercise, the phrase 'seventy divided by the product of seven and a number' is simplified by understanding that 'divided by' means /, and 'the product of' implies multiplication (*). Recognizing these operations allows students to accurately translate verbal statements into solvable problems.

Algebraic Expressions

In algebra, an algebraic expression is a combination of numbers, variables, and mathematical operations that represents a particular value or set of values. Variables, such as 'a number p' in the given verbal sentence, are symbols that stand in for unknown values. When we talk about the 'product of seven and a number p', in algebraic terms, we represent it as the expression '7 * p'.

The importance of correctly forming algebraic expressions cannot be overstated because they are the backbone of solving equations or inequalities. Instructors emphasize mastering this skill as it's integral to progressing in algebra and more advanced mathematics.

Problem-Solving in Algebra

In algebra, problem-solving often involves finding the unknown values of variables that make the equation true. Solving the equation demands a methodical approach, starting with the correct translation of a problem stated in words into a mathematical equation. Once the equation is established, various techniques such as simplifying expressions, isolating variables, and using inverse operations are applied to find the solution.

The essential step in problem-solving is first and foremost setting up the equation correctly. In our case, by translating the verbal sentence into the equation \( 70 / (7 \/cdot p) = 1 \), we provide a clear starting point for solving for 'p'. From there, we would apply appropriate algebraic principles and methods to find the value of the unknown variable.