The tens digit of a two-digit positive integer is 2 more...

The tens digit of a two-digit positive integer is 2 more than three times the units digit. If the digits are interchanged, the new number is 13 less than half the given number. Find the given integer. (Hint. Let $x=$ the tens-place digit and $y=$ the units-place digit; then $10 x+y$ is the number.

Key Concepts

Solution Verification in Context

After obtaining a solution through algebraic manipulation, it is important to substitute the values back into the original conditions to ensure they are met. This step confirms that the solution is not only mathematically consistent but also appropriate within the given problem context.

Digit Reversal and Numerical Manipulation

This concept examines the effects of interchanging the positions of digits in a number. For a two-digit number, reversing the digits changes its value from 10 times the tens digit plus the units digit to 10 times the units digit plus the tens digit. Understanding this reversal is key to formulating and solving problems where the reversed number bears a specific relationship to the original.

Systems of Linear Equations

This concept involves setting up and solving multiple linear equations simultaneously. When a problem provides more than one condition relating to the digits of a number, these conditions can be expressed as a system of equations. Solving the system yields values for the unknowns that satisfy all the conditions.

Algebraic Representation of Digits

This concept involves expressing the individual digits of a number with variables. In a two-digit number, the tens digit can be represented by one variable and the units digit by another, with the overall number written as 10 times the tens digit plus the units digit. This method is fundamental for forming algebraic equations based on the properties of the digits.

Translating Word Problems into Equations

This concept focuses on converting the narrative of a problem into algebraic expressions or equations. By deciphering the conditions given in words—such as relationships between digits or operations performed on the number—one can set up equations that model the problem mathematically.

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