Problem 20 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of twice a number and six is 14 . Find the number.

Step-by-Step Solution

Answer

The number is 4.

Step 1 - Define the variable

Let's define the unknown number as \( x \).

Step 2 - Translate the problem into an equation

The problem states that the sum of twice a number and six is 14. This can be written as an equation: \( 2x + 6 = 14 \).

Step 3 - Isolate the variable

Subtract 6 from both sides of the equation to isolate the term with the variable: \( 2x + 6 - 6 = 14 - 6 \), which simplifies to \( 2x = 8 \).

Step 4 - Solve for the variable

Divide both sides of the equation by 2 to solve for \( x \): \( \frac{2x}{2} = \frac{8}{2} \), which simplifies to \( x = 4 \).

Step 5 - Verify the solution

Substitute \( x = 4 \) back into the original word problem to check: Twice the number is \( 2 \times 4 = 8 \). Adding 6 to 8 gives \( 8 + 6 = 14 \). The solution is verified.

Key Concepts

Defining Variables

The first thing to do when solving an algebra word problem is to identify what the unknown is. In this case, the unknown number is what we are trying to find.
We call this creating a variable. A variable is a symbol, like x, that holds the place of the number we don't know yet.
Starting by defining the variable makes it easier to write equations and find the solution. We defined our variable as follows:

• Let x represent the unknown number.

Always choose simple and clear symbols to represent your unknowns to avoid confusion.

Writing Equations

Once we have a variable, we can translate the word problem into an algebraic equation. The problem states that the sum of twice a number and six equals 14.
This description converts directly into a mathematical expression:

• Twice the number: 2x
• Sum of twice the number and six: 2x + 6
• Equals 14: 2x + 6 = 14

This step is crucial because it turns the word problem into something we can solve mathematically.

Isolating Variables

To find the value of the variable, we need to isolate it on one side of the equation. This process involves several algebraic steps:

• Start with the equation: 2x + 6 = 14
• Subtract 6 from both sides to get: 2x = 8

Isolating the variable makes it straightforward to solve for x. You can think of isolating the variable as peeling away layers to get to the core value. Simplifying the equation step by step helps maintain accuracy and clarity.

Verifying Solutions

The final part of solving an algebra word problem is to verify that your solution makes sense. We do this by substituting the variable back into the original problem to see if the equation holds true.
In our problem, we found x = 4. We substitute 4 back into the original wording:

• Twice the number: 2 × 4 = 8
• Sum of twice the number and six: 8 + 6 = 14
• Indeed, 14 = 14, so our solution is correct.

Verifying ensures that no mistakes were made in calculations and that the solution is indeed valid. Double-checking your work is always a good practice to develop.