Problem 67 - Algebra for College Students

Question

Solve each problem by using a system of equations. The sum of the digits of a two-digit number is 7 . If the digits are reversed, the newly formed number is 9 larger than the original number. Find the original number.

Step-by-Step Solution

Answer

The original number is 34.

Step 1: Define Variables

Let the tens digit of the original number be represented by \( x \) and the units digit be represented by \( y \). This means the original number can be expressed as \( 10x + y \).

Step 2: Set Up the First Equation

According to the problem, the sum of the digits is 7. Therefore, the equation for this condition is:\[ x + y = 7 \]

Step 3: Set Up the Second Equation

When the digits are reversed, the number becomes \( 10y + x \). This new number is 9 larger than the original number, so we write the equation as:\[ 10y + x = 10x + y + 9 \]

Step 4: Simplify the Second Equation

Rearrange and simplify the second equation:\[ 10y + x = 10x + y + 9 \]\[ 10y - y + x - 10x = 9 \]\[ 9y - 9x = 9 \]

Step 5: Solve the System of Equations

We have the system of equations:1. \( x + y = 7 \)2. \( 9y - 9x = 9 \) (which simplifies to \( y - x = 1 \))Solve for \( y \) in terms of \( x \) using the second equation:\[ y = x + 1 \]

Step 6: Substitute and Solve for \( x \)

Substitute \( y = x + 1 \) into the first equation:\[ x + (x + 1) = 7 \]\[ 2x + 1 = 7 \]Subtract 1 from both sides:\[ 2x = 6 \]Divide by 2:\[ x = 3 \]

Step 7: Solve for \( y \)

Substitute \( x = 3 \) back into the equation \( y = x + 1 \):\[ y = 3 + 1 \]\[ y = 4 \]

Step 8: Find the Original Number

With \( x = 3 \) and \( y = 4 \), the original number is \( 10x + y = 10(3) + 4 = 34 \).

Key Concepts

Two-Digit Numbers

Two-digit numbers are composed of two separate parts: a tens digit and a units (or ones) digit. For example, in the number 34, 3 is the tens digit, and 4 is the units digit. Understanding this is crucial because a two-digit number is typically represented mathematically as \( 10x + y \), where \( x \) is the tens digit, and \( y \) is the units digit. This form is used because the tens digit is multiplied by 10 while the units digit is added directly. By using this representation, we can manipulate and solve problems involving two-digit numbers effectively.

Reversing Digits

Reversing the digits of a number means swapping the positions of the tens and units digits. For instance, if we have a number 34, reversing the digits gives us 43. The mathematical representation of this reversal for a number expressed as \( 10x + y \) would be \( 10y + x \). This concept is often used in problems that compare original numbers with their reversed counterparts, as seen in the provided exercise. By understanding how to represent a reversed number, you can set up equations to explore the relationships between the original and reversed numbers.

Digit Sum Problems

Digit sum problems involve conditions based on the sum of a number's digits. In the provided exercise, the sum of the digits of the original number is given as 7. This can be expressed algebraically as \( x + y = 7 \). In digit sum problems, knowing this sum helps in setting up one of the critical equations needed to solve the problem. These problems often require understanding both the structure and relationships of the digits that compose the number. By expressing these conditions mathematically, you gain a clear path to the solution.

Algebraic Problem Solving

Algebraic problem solving involves forming and solving equations to find unknown values. In the context of this exercise, we form a system of equations based on the problem's conditions. The first equation comes from the sum of the digits, \( x + y = 7 \), and the second comes from the comparison between the original and reversed numbers, \( y - x = 1 \). Solving these equations involves substitution or elimination methods. In this exercise, we substitute \( y = x + 1 \) into the first equation to solve for \( x \), and then use the result to find \( y \). This methodical approach is key to successfully solving problems involving relationships between numbers.