Problem 184 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is twenty-five. One number is five less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 15 and 10.

Step 1: Define the variables

Let the two numbers be represented by variables. Suppose the first number is denoted by \( x \) and the second number by \( y. \).

Step 2: Set up the first equation

According to the problem, the sum of the two numbers is twenty-five. Thus, the first equation is \( x + y = 25 \).

Step 3: Set up the second equation

The problem states that one number is five less than the other. To express this relationship, the second equation is \( y = x - 5 \).

Step 4: Substitute and solve for the first variable

Substitute the second equation into the first equation: \( x + (x - 5) = 25 \). Simplify and solve for \( x: 2x - 5 = 25 \). Adding 5 to both sides, \( 2x = 30 \). Dividing by 2, \( x = 15 \). Hence, the first number is 15.

Step 5: Find the second number

Substitute \( x = 15 \) back into the second equation \( y = x - 5 \) to find \( y: y = 15 - 5 = 10 \). Hence, the second number is 10.

Step 6: Verify the solution

To ensure correctness, check if the sum of 15 and 10 is 25 and whether one number is truly five less than the other. \( 15 + 10 = 25 \) and \( 15 - 10 = 5 \). Both conditions hold true. Therefore, the solution is correct.

Key Concepts

Defining Variables

When approaching a word problem in mathematics, it's essential to start by clearly defining the variables. Variables are symbols that represent unknown values, and in this case, we need two of them. Let's say the first number is represented by the variable \( x \), and the second number is represented by \( y \). Always choose variables that make sense to you. This step is crucial because it forms the foundation for writing your equations.

Solving Linear Equations

Once the variables are defined, it's time to set up the equations based on the problem's conditions. Linear equations are equations of the first order, meaning they involve only the first power of the variable. In our problem, the conditions given are:
1. The sum of two numbers is 25.
2. One number is five less than the other.
From the first condition, we can write the equation \( x + y = 25 \). From the second condition, we can write the equation \( y = x - 5 \). These two linear equations will help us find the values of \( x \) and \( y \).

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Here's how it works for our problem:
Start with the second equation \( y = x - 5 \). Substitute this equation into the first equation:

\( x + (x - 5) = 25 \)

Simplify it to get:

\( 2x - 5 = 25 \)

Add 5 to both sides:

\( 2x = 30 \)

Divide by 2:

\( x = 15 \).

Now that we have \( x \), substitute it back into the second equation:
\( y = 15 - 5 \).

This gives us \( y = 10 \). So, the numbers are 15 and 10.

Verification of Solution

Verifying your solution is an important final step. It ensures that your answers are correct. For our problem, we need to check two things:

1. The sum of the numbers is 25.
2. One number is five less than the other.

Substituting our solutions back into the original conditions:
1. \( 15 + 10 = 25 \)
2. \( 15 - 10 = 5 \).
Both conditions are satisfied, confirming our solution is correct. Always remember, verification gives you confidence in your final answer.