Problem 17 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The number is 15.

Step 1: Understand the Problem

Identify what is given and what needs to be found. Here, it is given that the difference between a number and 12 is 3, and we need to find the number.

Step 2: Formulate the Equation

Let the unknown number be represented by the variable . The problem states that the difference between and 12 is 3. This can be written as the equation: - 12 = 3.

Step 3: Solve the Equation

To find , add 12 to both sides of the equation: - 12 + 12 = 3 + 12. Simplifying this, = 15.

Step 4: Verify the Solution

Substitute = 15 back into the original equation to ensure it satisfies the condition: 15 - 12 = 3, which confirms the solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use variables representing unknown values. These equations often involve operations like addition, subtraction, multiplication, and division.
In our exercise, the statement 'The difference of a number and 12 is three' can be transformed into an algebraic equation.
Here, the unknown number is represented by a variable, which we'll call \( x \).
So, the algebraic equation becomes:

\( x - 12 = 3 \).
This equation allows us to use algebraic techniques to find the value of the unknown variable \( x \) and solve for it.

Problem-Solving Steps

Solving a number word problem can be broken down into clear steps:

• Understand the Problem
• Formulate the Equation
• Solve the Equation
• Verify the Solution

Let's break these steps down based on our exercise.
First, identify what is given (the difference of a number and 12 is three) and what needs to be found (the number).
Next, create an equation based on the given information. Then, use algebra to find the value of the variable.
Finally, verify that the solution we found is indeed correct by substituting it back into the original equation.

Variable Representation

In algebra, variables are symbols that represent unknown values. Common symbols include \( x \), \( y \), and \( z \).
In this exercise, we let \( x \) represent the unknown number.
When we see a phrase like 'the difference of a number and 12 is three,' we translate it into an equation using our variable as follows:

\( x - 12 = 3 \).
This representation provides a clear and concise way to set up our problem so that we can solve it using algebraic methods.
Choosing an appropriate variable and correctly translating words into equations is crucial for solving word problems.

Equation Verification

After solving the equation, it is essential to verify that our solution is correct.
We do this by substituting the solution back into the original equation.
In our solved problem, we found \( x = 15 \). So, we substitute 15 back into the equation \( x - 12 = 3 \):

\( 15 - 12 = 3 \).
This confirms that our solution is correct since both sides of the equation are equal.
Verification is a crucial final step to ensure accuracy and confirm that the solution meets all the conditions described in the problem.