Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Five less than 3 times a number gives 7

Step-by-Step Solution

Answer

The correct equation is \(3x - 5 = 7\)

Step 1: Identify the Variable

Given that \(x\) is the variable which represents 'the number' in the statement.

Step 2: Translate the Sentence to Mathematical Expression

The phrase '3 times a number' translates to 3 multiplied by \(x\) or \(3x\). And 'Five less than 3 times a number' means we subtract 5 from \(3x\), which translates to \(3x - 5 \).

Step 3: Construct the Equation

Finally, 'gives 7' means the result is 7. Therefore, we equate \(3x-5\) to 7. The final equation is: \(3x - 5 = 7\) .

Key Concepts

Variable Representation

In many word problems, we need to choose a variable to represent an unknown number or quantity. A variable is like an empty box we can fill with the correct answer once we figure it out. In this case, we are asked to let the variable \(x\) represent the number.

This step is crucial because it provides a foundation for the rest of the problem. When you see a phrase like "a number," you should immediately think about using a variable. Here are some tips for choosing your variable:

• Stick with a common letter like \(x\) when not otherwise specified.
• Be consistent in your variable choice throughout the problem.
• Remember that the variable is a placeholder for the unknown value you will solve for.

Using \(x\), we simplify our understanding of the problem and set ourselves up for the next steps.

Mathematical Expressions

After choosing a variable, the next step is translating parts of the word problem into mathematical expressions. Let's break down the problem: "Five less than 3 times a number gives 7."

The phrase "3 times a number" means we multiply the variable \(x\) by 3, which gives us the expression \(3x\). This expression represents "3 times the number" mathematically.

Next, "five less than 3 times a number" requires us to perform an operation. This involves subtraction. We take \(3x\) and subtract 5 to form the expression \(3x - 5\).

Here's a quick guide to translating phrases into expressions:

• "More than" or "less than" indicates addition or subtraction.
• "Times" or "product" suggests multiplication.
• "Divided by" refers to division.

Remember, mathematical expressions are smaller pieces of the larger equation we aim to construct.

Equation Construction

The culmination of translating a word problem is forming a complete equation. An equation expresses a statement of equality between two expressions. The problem states, "Five less than 3 times a number gives 7."

At this stage, we already have the mathematical expression \(3x - 5\) from the previous step. To construct the equation, we look at "gives 7." This tells us that the expression results in or equals 7. Hence, we set our expression equal to 7, forming the equation:

\[3x - 5 = 7\]

Equation construction requires gathering all pieces and ensuring the mathematical representation matches the verbal description accurately. Some tips include:

• Ensure both sides of the equation are balanced.
• Use the equals sign to denote the relationship between the expressions.
• Consider re-checking the word problem to ensure all expressions are included in the equation.

With your equation correctly constructed, you're ready to solve for the variable \(x\).