Problem 65 - Algebra for College Students

Question

Solve each problem by using a system of equations. The tens digit of a two-digit number is 1 more than three times the units digit. If the sum of the digits is 9 , find the number.

Step-by-Step Solution

Answer

The number is 72.

Step 1: Understand the problem

We are given a two-digit number where the tens digit is 1 more than three times the units digit. Additionally, the sum of the digits is 9. We need to find the number.

Step 2: Define variables

Let the units digit be denoted by \( x \) and the tens digit by \( y \).

Step 3: Formulate the first equation

According to the first part of the problem, the tens digit is 1 more than three times the units digit. Translated into an equation: \( y = 3x + 1 \).

Step 4: Formulate the second equation

The sum of the digits equals 9. This can be expressed as: \( x + y = 9 \).

Step 5: Substitute and solve

Substitute \( y = 3x + 1 \) from the first equation into the second equation: \[ x + (3x + 1) = 9 \]Simplify and solve for \( x \):\[ 4x + 1 = 9 \] \Subtract 1 from both sides: \\[ 4x = 8 \] \Divide by 4: \\[ x = 2 \]

Step 6: Find the tens digit

Using \( x = 2 \), substitute back into the equation for \( y \): \[ y = 3(2) + 1 = 7 \]. Thus, the tens digit is 7.

Step 7: Determine the number

The number is formed by the tens digit and the units digit, which are 7 and 2 respectively, making the number 72.

Key Concepts

two-digit number

A two-digit number is any number that ranges from 10 to 99, inclusive. These numbers are made up of two distinct parts: the tens digit and the units (or ones) digit. For example, in the number 34, the tens digit is 3 and the units digit is 4. Understanding the structure of a two-digit number is critical when working with digit-related problems, as it helps in formulating the equations needed to solve these problems.

• The tens digit represents how many groups of ten are present in the number.
• The units digit indicates the remaining part that, when added to the tens, makes the complete number.

Working with two-digit numbers often involves examining the relationships between these two digits to solve mathematical puzzles.

equation formulation

Equation formulation is a crucial step in solving math problems, especially when dealing with digit problems. It involves translating the given conditions of a problem into mathematical equations that can be solved.

In the given problem, we identify two key conditions:

• The tens digit is 1 more than three times the units digit. Mathematically, this becomes the equation: \( y = 3x + 1 \).
• The sum of the digits equals 9, which translates to: \( x + y = 9 \).

Here, \( x \) represents the unit digit, and \( y \) represents the tens digit. Formulating these equations allows us to use algebraic methods, such as substitution and elimination, to solve for the unknown digits.

digit problems

Digit problems can seem daunting at first, but they usually follow a specific pattern or relationship between the digits that can be solved step-by-step. These problems often ask you to determine the digit values or sum based on given conditions.

The challenge lies in translating the words into equations. For instance, in our example problem:

• "The tens digit is 1 more than three times the units digit" directly translates into an algebraic expression.
• Understanding the relationships expressed in words helps form the foundation for solving the problem using systems of equations.

Tackling digit problems requires careful attention to detail and practice in forming representative equations.

problem-solving steps

Effective problem-solving involves a systematic approach, especially with systems of equations. This structured methodology ensures that each condition of the problem is addressed and solvable through algebraic techniques. Here, these steps are crucial:

• **Understanding the Problem:** Read carefully and identify what is being asked.
• **Define Variables:** Assign symbols to the unknowns, such as \( x \) and \( y \) for digits.
• **Formulate Equations:** Based on the problem, create mathematical equations that represent the conditions.
• **Substitute and Solve:** Use substitution methods to find a solution for the variables.
• **Verification:** Once solved, double-check by substituting back to ensure the solution satisfies the original conditions.

Following each of these steps methodically not only improves accuracy but also helps develop a clearer understanding of how different mathematical concepts are applied.