Problem 15 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of a number and eight is 12 . Find the number.

Step-by-Step Solution

Answer

The number is 4.

Step 1 - Define the variable

Let the unknown number be denoted by the variable, say, \( x \).

Step 2 - Write an equation

According to the problem, the sum of the number and eight is 12. This can be expressed as the equation:\[ x + 8 = 12 \]

Step 3 - Solve for the variable

To solve for \( x \), subtract 8 from both sides of the equation:\[ x + 8 - 8 = 12 - 8 \] Simplifying this, we get: \[ x = 4 \]

Step 4 - Verify the solution

To ensure the solution is correct, substitute \( x = 4 \) back into the original equation:\[ 4 + 8 = 12 \] Since the left-hand side equals the right-hand side of the equation, the solution is verified.

Key Concepts

solving equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. Here's a simple process to solve equations:

For the equation given: \[ x + 8 = 12 \]

• First, identify the equation that represents the problem. In this case, the sum of a number and eight equals 12.
• To isolate the variable, perform the inverse operation on both sides. Here, we subtract 8 from both sides to cancel out the +8.
• This gives us: \[ x = 4 \]

Breaking down each step helps make the process clear and methodical. Always remember to do the same operation on both sides to keep the equation balanced.

defining variables

Variables are symbols used to represent unknown values. In word problems, identifying and defining variables is crucial for setting up equations.

Here's how to define variables effectively:

• Read the problem carefully to understand what you need to find.
• Choose a letter, commonly x, to represent the unknown number.
• In the given exercise, let the unknown number be x.

Defining the variable sets the foundation for creating an equation, making the problem easier to solve. It helps to write down the meaning of the variable to avoid confusion later.

verification of solutions

Verification is the step where you check your solution to ensure it's correct. This involves substituting your answer back into the original equation.

For our specific problem:

• Solve the equation and find that \[ x = 4 \]
• To verify, substitute 4 back into the original equation \[ x + 8 = 12 \]
• Substituting gives \[ 4 + 8 = 12 \]
• The left-hand side equals the right-hand side, confirming the solution is correct.

Verification builds confidence in your solution and is a good habit to develop in all algebra problems. It ensures you didn't make any mistakes in your calculations.