Problem 68 - Algebra for College Students

Question

Solve each problem by using a system of equations. The units digit of a two-digit number is 1 less than twice the tens digit. If the digits are reversed, the newly formed number is 27 larger than the original number. Find the original number.

Step-by-Step Solution

Answer

The original number is 47.

Step 1: Define the Variables

Let's define the variables. Let \( x \) be the tens digit and \( y \) be the units digit of the original number. So, the original number can be expressed as \( 10x + y \).

Step 2: Formulate the First Equation

According to the problem, the units digit \( y \) is 1 less than twice the tens digit \( x \). Therefore, we have the equation: \ y = 2x - 1 \

Step 3: Formulate the Second Equation

When the digits are reversed, the new number is \( 10y + x \), and this number is 27 more than the original number. Therefore, we have the equation: \ 10y + x = 10x + y + 27 \

Step 4: Simplify the Second Equation

Simplify the second equation to solve for one of the variables: \ 10y + x = 10x + y + 27 \Subtract \( y \) and \( x \) from both sides: \ 9y = 9x + 27 \Divide the entire equation by 9 to further simplify: \ y = x + 3 \

Step 5: Solve the System of Equations

Now you have two equations: \ y = 2x - 1 \ y = x + 3 \Set \( 2x - 1 = x + 3 \) to solve for \( x \): \ 2x - x = 3 + 1 \ x = 4 \ Now substitute \( x = 4 \) into \( y = x + 3 \): \ y = 4 + 3 = 7 \

Step 6: Find the Original Number

Now that we have \( x = 4 \) and \( y = 7 \), the original number is formed as \( 10x + y = 10(4) + 7 = 47 \).

Key Concepts

Two-Digit Numbers

A two-digit number is a number that consists of two digits, typically ranging from 10 to 99. In our exercise, we are considering a number with a tens digit and a units digit. The tens digit is the first digit from the left, while the units digit is the second digit.
For example, in the number 47:

• The '4' is the tens digit.
• The '7' is the units digit.

Understanding the concept of a two-digit number is crucial because it lays the foundation for setting up our equations. We can represent such a number in algebra using the formula: \[ \text{Number} = 10 \times (\text{tens digit}) + (\text{units digit}) \]By assigning variables to these digits, we convert the number into an algebraic expression, which offers a significant advantage when solving complex problems.

Digit Reversal

Digit reversal refers to changing a number by swapping its tens digit and units digit. This changes the value of the original number, and it's a handy technique for math problems involving number manipulation.
In our exercise, the original two-digit number is represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit. When we reverse these digits, the number becomes \(10y + x\).
This reversal is not just about shuffling digits; it significantly alters the number's value and is used to form a new equation in our system of equations. The problem states that this reversed number should be larger by 27, providing us with the basis for one of our solutions.

Variable Definition

Defining variables is the first step in translating a word problem into a mathematical problem. It involves assigning symbols or letters to represent unknown quantities. In this exercise:

• Let \( x \) be the tens digit of the original number.
• Let \( y \) be the units digit of the original number.

Using these variables, we can express the original number as \( 10x + y \), allowing us to create equations based on the problem's conditions.
Variable definition helps in constructing expressions or equations that represent real-world scenarios. It is essential as it translates the problem into a form that can be manipulated algebraically to find the solution.

Algebraic Equations

Algebraic equations are mathematical statements indicating that two expressions are equal. In our problem, we used algebraic equations to represent different conditions given.First, the problem states that the units digit is 1 less than twice the tens digit, leading to the equation:\[ y = 2x - 1 \]
Next, it tells us that reversing the digits results in a number that is 27 more than the original, leading to the equation:\[ 10y + x = 10x + y + 27 \]
By simplifying these equations, we get:

• \( y = x + 3 \)
• \( y = 2x - 1 \)

These are linear equations, and solving them simultaneously involves substitution or elimination strategies, leading to the solution for \( x \) and \( y \). Using these methods allows us to find the values of the digits and thus the original number.