Problem 33 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When seven times a number is decreased by two times the number, the result is negative one.

Step-by-Step Solution

Answer

The equation is \( 5x = -1 \).

Step 1: Identify the Unknown Variable

Let's denote the unknown number as \( x \). This will allow us to express the relationship described in the sentence mathematically.

Step 2: Translate the Phrase to an Equation

The phrase "seven times a number" translates to \( 7x \), and "two times the number" translates to \( 2x \). The phrase "decreased by" suggests a subtraction. So the expression becomes \( 7x - 2x \).

Step 3: Express the Result

The phrase "the result is negative one" translates to setting the expression equal to \(-1\). Hence, the equation becomes \( 7x - 2x = -1 \).

Step 4: Simplify the Equation

Combine like terms in the equation: \( 7x - 2x = 5x \). The equation simplifies to \( 5x = -1 \).

Step 5: The Final Equation

The mathematical expression derived from the phrase is \( 5x = -1 \), which represents the sentence.

Key Concepts

Understanding the Unknown Variable

In algebra, an unknown variable is typically represented by a symbol such as \( x \). This symbol stands for a number that we do not know yet, or are trying to find in a problem. Whenever you come across phrases such as "a number," it often indicates the presence of an unknown variable.

Using an unknown variable allows us to set up equations and solve them to find the value of this number. For example, in the phrase "when seven times a number is decreased by two times the number," "a number" refers to an unknown quantity which we express by \( x \).

• It provides a way of expressing conditions or relationships in a simplified form.
• Representing unknowns with variables helps track the problem components.

This stepping stone enables us to move towards forming mathematical expressions, ultimately leading to equations that reveal the unknown number's value.

Crafting a Mathematical Expression

A mathematical expression is a combination of numbers, variables, and mathematical operators (like addition or subtraction) that represent a value or a relationship. In our exercise, we translate the given words into a mathematical form as follows:

• "Seven times a number" becomes \( 7x \).
• "Decreased by two times the number" implies subtracting \( 2x \).

Putting it together, the expression "seven times a number decreased by two times the number" is converted into \( 7x - 2x \).

These expressions serve as the building blocks in constructing an equation. Understanding how to translate verbal descriptions into mathematical expressions is crucial:

• It sets the groundwork for further mathematical computation.
• Accurately capturing the relationships described verbally in mathematical terms is essential for solving equations.

Formulating an Algebraic Equation

An algebraic equation is a mathematical statement where two expressions are set equal to each other. It acts as a bridge, guiding us to find the unknown variable. In this step of our problem, we move from a mere expression to a complete algebraic equation by interpreting the phrase "the result is negative one."

Starting with our expression \( 7x - 2x \), we establish the equation by equating it to \(-1\), yielding the equation \( 7x - 2x = -1 \). This equation now conveys the complete relationship defined in the verbal problem.

• Equations require balancing, where both sides represent the same value.
• This places us in a position to solve for \( x \), providing the missing numerical value.

The simplification of this equation (combining like terms to transform it into \( 5x = -1 \)) makes it more manageable to solve. Mastering algebraic equations is vital as they are widely applied to solve real-world problems.