Problem 71 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. The product of 4 and a number increased by 5 is 33

Step-by-Step Solution

Answer

The translated equation is \(4x + 5 = 33\).

Step 1: Identify the Variable

In the problem, it's mentioned that let 'x' represent the number. So, we'll denote the unknown number in the problem by the variable 'x'.

Step 2: Translate 'product of 4 and a number'

'The product of 4 and a number' means 4 times the number. So, this can be written as \(4x\).

Step 3: Translate 'increased by 5'

'Increased by 5' means adding 5. So, we will add 5 to the product of 4 and the number. This can be written as \(4x + 5\).

Step 4: Translate 'is 33'

'Is 33' signifies equality with 33. The equals sign in a mathematical equation represents the word 'is'. Thus, we will make the expression \(4x + 5\) equal to 33. This can be written as \(4x + 5 = 33\).

Step 5: Write the Complete Equation

Now we write the whole equation according to the translation. The sentence 'The product of 4 and a number increased by 5 is 33' translates into the equation \(4x + 5 = 33\). This is the final equation.

Key Concepts

Variable Identification

When tackling algebraic equations, one of the first steps is to identify the variable. A **variable** is a symbol, often a letter, used to represent an unknown value in mathematics. In this exercise, the variable is given as "the number" we need to find. Here, we use the letter \( x \) to represent this unknown number.

This process establishes what you are trying to discover or solve for. You can think of a variable as a placeholder for the missing piece of an equation's puzzle.

When identifying variables, follow these steps:

• Read the problem carefully to determine what you're solving for.
• Select a letter, commonly \( x \), \( y \), or \( z \), to serve as your variable.
• Ensure your variable accurately reflects the unknown value you need to find.

By naming the variable, you're ready to move on to translating the verbal description into a mathematical equation.

Equation Translation

Once the variable is set, the next step is **equation translation**. This process involves turning a description or sentence into a mathematical equation.

For instance, in the sentence "The product of 4 and a number increased by 5 is 33," we break it down as follows:

• The phrase "product of 4 and a number" refers to the multiplication 4 times the variable \( x \), which is written as \( 4x \).
• The term "increased by 5" means you'll add 5 to the current product, giving us \( 4x + 5 \).
• The phrase "is 33" indicates the result or outcome of the calculation, expressed by the equals sign \( = 33 \).

Thus, the full equation reads \( 4x + 5 = 33 \).

The skill of translating words into symbols is crucial, as it allows you to work mathematically with real-world situations.

Solving Linear Equations

After translating the sentence into an equation, the task becomes one of **solving linear equations**. A linear equation is an equation in which the highest power of the variable is one.

With the equation \( 4x + 5 = 33 \), our goal is to find the value of \( x \) that makes the equation true. Here’s a straightforward approach:

• Subtract 5 from both sides to isolate the term with the variable: \( 4x + 5 - 5 = 33 - 5 \) simplifies to \( 4x = 28 \).
• Divide both sides by 4 to solve for \( x \): \( \frac{4x}{4} = \frac{28}{4} \), which simplifies to \( x = 7 \).

Solving equations involves operations that maintain balance, just like balancing a scale. By isolating the variable \( x \), you find its value and solve the problem.

Mastering linear equations opens the door to more complex mathematics, as you gain confidence in tackling various problem types.