Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. The product of 6 and a number increased by 3 is 33 .

Step-by-Step Solution

Answer

The sentence translates to the equation as \(6x + 3 = 33\).

Step 1: Identify the variable

A number in the sentence is represented by the variable \(x\).

Step 2: Translate the sentence into an equation

'The product of 6 and a number' translates to '6 times \(x\)', or \(6x\). 'Increased by 3' translates to 'plus 3', or \(+3\). 'is' translates to equals, or \(=\). Lastly, '33' stays as \(33\). Putting all of this together results in \(6x + 3 = 33\).

Key Concepts

Translation of Sentences to Equations

In algebra, translating sentences into equations is like decoding a message. This process involves converting a problem stated in words into mathematical language. Let’s break down how this works using the provided example sentence: "The product of 6 and a number increased by 3 is 33."

First, identify keywords:

• "Product" means multiplication.
• "Increased by" refers to addition.
• "Is" signals equality, like an equal sign.

Let’s translate step-by-step:

- "The product of 6 and a number" means 6 times a number, which we express as \(6x\).
- "Increased by 3" implies we add 3, written as \(+3\).
- "Is 33" simply means equals 33, or \(= 33\).

Putting it all together creates the equation \(6x + 3 = 33\). The translation process is crucial in solving word problems algebraically as it forms the foundation for identifying relationships between values.

Variables in Algebra

Variables are fundamental elements in algebra, essentially acting as placeholders for numbers we seek to find. Think of them as boxes that can hold any number. In our problem, the letter \(x\) is used as the variable representing a specific number.

Key points about variables:

• Variables allow us to create general expressions that apply to many situations.
• They are often represented by letters, with \(x\) being the most common choice.
• In equations, they are the unknowns you solve for.

In the sentence "The product of 6 and a number," the phrase "a number" was the unknown. We substituted it with \(x\), forming the expression \(6x + 3 = 33\).

Understanding variables helps us manipulate and solve mathematical equations effectively. They are the bridge between the physical world and abstract mathematical representations, crucial for handling real-life problems.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Let’s use the example equation \(6x + 3 = 33\) and solve it.

• First, isolate terms involving \(x\) on one side by removing the constant term. Subtract 3 from both sides:
  \(6x + 3 - 3 = 33 - 3\) simplifies to \(6x = 30\).
• Next, solve for \(x\) by dividing both sides by 6:
  \(\frac{6x}{6} = \frac{30}{6}\), simplifying to \(x = 5\).

Through these steps, we determined that \(x = 5\). Solving linear equations is a systematic process:

- Simplify both sides of the equation if necessary.
- Perform inverse operations to isolate the variable.
- Always check the solution by substituting back into the original equation, confirming that it holds true.