Problem 58 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. The sum of two numbers is 103 . The larger number is one more than five times the smaller number. Find the numbers.

Step-by-Step Solution

Answer

The smaller number is 17 and the larger number is 86.

Step 1: Define Variables

Let the smaller number be represented by \( x \). Since the larger number is one more than five times the smaller number, represent the larger number as \( 5x + 1 \).

Step 2: Set Up an Equation

According to the problem, the sum of the two numbers is 103. Therefore, we can write the equation \( x + (5x + 1) = 103 \).

Step 3: Simplify the Equation

Simplify the equation by combining like terms: \( x + 5x + 1 = 103 \) simplifies to \( 6x + 1 = 103 \).

Step 4: Solve for x

Subtract 1 from both sides to isolate the term with \( x \): \( 6x = 102 \). Then, divide by 6 to find \( x \): \( x = 17 \).

Step 5: Find the Larger Number

Now that you know \( x = 17 \), substitute it back into the expression for the larger number: \( 5x + 1 = 5(17) + 1 = 85 + 1 = 86 \).

Step 6: Verify the Solution

Check the solution by ensuring the sum of the numbers is 103: \( 17 + 86 = 103 \), which matches the problem statement. Thus, the solution is verified.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. In our exercise, we were given statements about two numbers and had to formulate these into an algebraic equation. This equation helps us find unknown values.
By converting word problems into equations, we can analyze and solve them systematically. For instance, the statement "the sum of two numbers is 103" becomes an equation showing the relationship between two unknowns.
Remember:

• Translate verbal descriptions into mathematical expressions.
• Identify which parts of the statement form the equation's left and right sides.
• Maintain balance by keeping both sides of the equation equal.

This practice not only aids in solving math problems but also hones problem-solving skills for real-world issues.

Defining Variables Effectively

Variable definition is crucial in simplifying complex problems, as it represents unknown quantities symbolically. In the exercise, we defined the smaller number as the variable \( x \). This is our starting point to express other related quantities.
The larger number was then defined uniquely as \( 5x + 1 \), since it was one more than five times the smaller number.
Key Points:

• Select clear and consistent variables to avoid confusion.
• Ensure each variable aligns with the problem's context and conditions.
• Organize all expressions involving variables on one side for simplicity.

Properly defining variables is like setting the foundation for a sturdy building in problem-solving.

The Process of Solving Linear Equations

Solving linear equations involves finding the variable values that make the equation true. In our problem, we began with the equation \( x + (5x + 1) = 103 \). We needed to isolate \( x \).
Steps:

• Combine like terms: Simplify the equation \( x + 5x + 1 = 103 \) to get \( 6x + 1 = 103 \).
• Isolate terms with \( x \): Subtract 1 from both sides, resulting in \( 6x = 102 \).
• Resolve \( x \): Divide by 6 to find that \( x = 17 \).

Each step peels back a layer of complexity by reducing terms or re-arranging, making it easier to understand and find solutions.

Verification of Solutions

Verification is the final step in problem-solving, confirming that the solution indeed meets the original problem's requirements. For our exercise, we recalculated the sum of the numbers to ensure they added up to 103.
Here's how:

• Calculate both numbers using your solution values: Smaller number is \( 17 \), and the larger is \( 86 \) (calculated from \( 5(17) + 1 \)).
• Check the sum: \( 17 + 86 \) indeed equals 103.
• Ensure all conditions from the problem description are met.

This step solidifies your problem-solving ability, confirming accuracy and understanding.