Problem 21 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The difference of twice a number and seven is 17 . Find the number.

Step-by-Step Solution

Answer

The number is 12.

Step 1: Define the variable

Let the unknown number be represented by the variable \( x \).

Step 2: Set up the equation

According to the problem, twice the number minus seven is equal to 17. This can be written as the equation: \[ 2x - 7 = 17 \]

Step 3: Isolate the variable term

Add 7 to both sides of the equation to isolate the term with the variable: \[ 2x - 7 + 7 = 17 + 7 \]which simplifies to: \[ 2x = 24 \]

Step 4: Solve for the variable

Divide both sides by 2 to solve for \( x \): \[ x = \frac{24}{2} \]which simplifies to: \[ x = 12 \]

Step 5: Verify the solution

Substitute \( x = 12 \) back into the original equation to ensure it satisfies the condition: \[ 2(12) - 7 = 24 - 7 = 17 \]Since the left-hand side equals the right-hand side, the solution is verified.

Key Concepts

Defining Variables

In algebra, it is important to represent unknown values with variables like \( x \) or \( y \). In this exercise, we are asked to find an unknown number. We start by defining this number as \( x \). This process is essential because it allows us to use algebraic methods to find the unknown value. By defining variables, we transform a word problem into a mathematical one that is easier to handle.

Setting Up Equations

After defining the variable, the next step is to set up an equation based on the given problem. The problem states that 'the difference of twice a number and seven is 17'. In mathematical terms, this can be written as:
\( 2x - 7 = 17 \)
Here, 'twice a number' translates to \( 2x \), and 'the difference of ... and seven' means we subtract 7.
Our goal is to establish a clear relationship between the numbers using equations. By setting up the correct equation, solving the problem becomes a straightforward process.

Solving Equations

To solve the equation \( 2x - 7 = 17 \), we need to isolate the variable \( x \). Follow these steps to solve it:

• Add 7 to both sides of the equation: \( 2x - 7 + 7 = 17 + 7 \). This simplifies to \( 2x = 24 \).
• Next, divide both sides by 2 to solve for \( x \): \( x = \frac{24}{2} \). This simplifies to \( x = 12 \).

Every step should be performed carefully to avoid errors. Each manipulation helps to move towards isolating \( x \) and finding its value.

Verification of Solution

After solving the equation, always verify your solution. Substituting \( x = 12 \) back into the original equation, we get:

• \( 2(12) - 7 = 24 - 7 = 17 \).

The left side of the equation equals the right side, confirming that our solution \( x = 12 \) is correct. Verification is a crucial step because it ensures the solution meets the conditions of the original problem.

Number Word Problems

Number word problems can seem challenging at first. But, by systematically transforming words into mathematical expressions, they become manageable.
The key steps are:

• Identify and define variables.
• Set up equations based on the relationships described in the problem.
• Solve the equations methodically.
• Verify the solutions to ensure accuracy.

Following these steps will help demystify number word problems, making them easier to understand and solve.