Problem 25 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. The product of 14 and a number \(x\) is one.

Step-by-Step Solution

Answer

The equation corresponding to the verbal sentence is \(14x = 1\).

Step 1 - Identify Operations

The keyword 'product' denotes multiplication in mathematics. When we say 'product of 14 and a number \(x\)', this implies that we need to multiply 14 with \(x\).

Step 2 - Identify The Relationships

The word 'is' refers to the '=' sign in a mathematical equation or inequality. That is, 'is one' indicates that the result of the multiplication operation equals one.

Step 3 - Write Equation

Combining the analysis from the previous steps, we can write a mathematical equation to sum up the verbal sentence: 'The product of 14 and a number \(x\) is one' as \(14x = 1\).

Key Concepts

Mathematical Operations

Understanding mathematical operations is like learning the basics of a language. It's about knowing how to add, subtract, multiply, and divide numbers. It also includes understanding more complex operations such as exponentiation and finding roots. For example, in our exercise, the term 'product' refers specifically to the result of multiplication. So when we're told to find the product of 14 and a number, we're being instructed to multiply those two values together. This fundamental knowledge is the building blocks for translating verbal sentences into equations.

Let's practice identifying operations: If you hear 'sum', think addition; for 'difference', think subtraction; 'product' implies multiplication; and 'quotient' refers to division. When you come across these terms, picturing the associated mathematical symbol can help you rapidly convert words into a numerical or algebraic expression.

Writing Equations

Writing equations is an essential skill in mathematics that bridges the gap between a verbal description of a problem and its numerical analysis. When you write an equation, you're creating a mathematical statement that shows the equality between two expressions. In the given exercise, once we know the operations and relationships, writing the equation is straightforward.

Here's a tip: Always start by determining what you're being asked to find, which is often represented by a variable like 'x' or 'y'. Next, translate the verbal clues into mathematical symbols and numbers, writing them as an equation. In this case, the verbal clue 'the product of 14 and a number is one' translates to the equation \(14x = 1\). Always remember to keep the equation balanced – what you do to one side of the equals sign, you must do to the other.

Understanding Math Vocabulary

Understanding math vocabulary is critical for successfully translating words into equations. Terms like 'product', 'is', and 'number' are part of this vocabulary. Each word tells you something about the mathematical operation or the structure of the equation you're about to write. 'Product' implies multiplication, while 'is' indicates equality, and 'number' suggests a variable or specific value is involved.

It's helpful to create a cheat sheet of key terms and their mathematical meanings. If you're just getting started, keep a dictionary or glossary of terms handy while working through problems. Sometimes a single word can change the entire meaning of a problem, so paying close attention to the vocabulary used is just as important as the numbers themselves.

Solving Algebraic Equations

Solving algebraic equations is the process of finding the value or values that satisfy the equation. Once you've successfully translated a verbal sentence into an equation, the next step is solving it. For simple equations, like the one in our exercise \(14x = 1\), the solution involves performing operations that will isolate the variable on one side. To solve this, you would divide both sides of the equation by 14, yielding \(x = \frac{1}{14}\).

More complex equations may require additional steps such as distributing, combining like terms, or using the quadratic formula. Notwithstanding the complexity, the principles remain the same: do the same operation to both sides of the equation to maintain balance and isolate the variable for which you're solving.