Problem 64 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. 3 is the quotient of a number and \(-6\)

Step-by-Step Solution

Answer

The equation is \(3 = \frac{x}{-6}\).

Step 1: Identify the Need for Variable

The problem states 'Let x represent the number'. So, x is the variable representing the unknown number.

Step 2: Understand Meaning of Quotient

The quotient of a number and another number is the result of the first number divided by the other number. So, 'quotient of a number and -6' means 'x divided by -6'.

Step 3: Form the Equation

The problem states '3 is the quotient of a number and -6'. Therefore, we can write this sentence as an equation: \(3 = \frac{x}{-6}\).

Key Concepts

Variable Representation

In math, variables are symbols that stand in for unknown numbers or values. They're helpful when we don't know the exact figure and use them to express mathematical relationships or rules. In our exercise, we use the variable \( x \) to represent an unknown number.

• Variables can be letters like \( x \), \( y \), or \( z \), but they serve the same purpose: to link unknown values to a mathematical operation.
• Choosing \( x \) as the variable makes it simpler to communicate the essence of the problem. It's like placing a placeholder in a sentence, indicating where the unknown figure is.

Think of variables as blank spaces you can fill with numbers to see how changes affect the equation. This blank space is crucial in translating a sentence into mathematical language.

Understanding Quotients

Quotients play a foundational role in division. When the term "quotient" is used, it refers to the result of one number being divided by another. In our case, the phrase "the quotient of a number and \(-6\)" translates directly to dividing the unknown number \( x \) by \(-6\).

• Consider a simple division: when you divide 10 by 2, the quotient is 5. Here, 10 is the number being divided, and 2 is the divisor.
• If the result of dividing one number by another is negative, as it is with \( -6 \), the division involves a negative divisor.

Understanding quotients helps in recognizing how words convert to numbers and symbols in equations. It highlights relationships, showing how parts come together in an equation.

Formulating Equations

Formulating equations from sentences is like translating a language. It's specifying a relationship between numbers using mathematical symbols. The goal is to capture the precise relationship described in words. In our example: "3 is the quotient of a number and \(-6\)" is expressed mathematically as an equation. Here, we're stating that dividing our unknown number \( x \) by \(-6\) results in 3. Hence, the equation:\[3 = \frac{x}{-6}\]

• The left side of the equation shows the result, which is \( 3 \).
• The right side expresses the operation or action, dividing \( x \) by \(-6\).

This equation captures the essence of the original sentence. Practicing the skill of equation formulation enhances problem-solving abilities. It allows you to decode real-world problems into understandable math, bridging concepts across topics.