Problem 21 - Elementary Algebra

Question

Set up an algebraic equation and then solve. If the smaller of two consecutive integers is subtracted from two times the larger, then the result is 17 . Find the integers.

Step-by-Step Solution

Answer

The integers are 15 and 16.

Step 1: Define the Variables

Let the smaller integer be \( x \). Then the larger consecutive integer is \( x+1 \).

Step 2: Set Up the Equation

According to the problem statement, if the smaller integer (\( x \)) is subtracted from two times the larger integer (\( 2(x + 1) \)), the result is 17. This can be set up as the equation:\[ 2(x + 1) - x = 17 \]

Step 3: Simplify the Equation

Distribute the 2 in the equation:\[ 2x + 2 - x = 17 \]Combine the terms to simplify:\[ x + 2 = 17 \]

Step 4: Solve the Equation for x

Subtract 2 from both sides to solve for \( x \):\[ x = 17 - 2 \]\[ x = 15 \]

Step 5: Determine the Consecutive Integers

The smaller integer is \( 15 \), and the larger consecutive integer is \( x + 1 = 16 \).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order. They have a difference of 1 between each pair of numbers. When dealing with problems involving consecutive integers, you first identify the sequence. For example, if the first integer is expressed as \( x \), the next consecutive integer will be \( x + 1 \). This pattern helps in setting up an algebraic equation that represents the relation described in the problem.

• The first integer: \( x \)
• The second (next) integer: \( x + 1 \)

Understanding how to express these integers algebraically sets the stage for accurately forming equations, which is essential in solving many types of mathematical problems.

Solving Equations

Solving equations is a key skill in algebra. The goal is to find the value of an unknown variable that makes the equation true. First, you create the equation based on the problem. For example, using the two integers mentioned, you set up an equation as follows:

• Start with the expression: \( 2(x + 1) - x = 17 \).
• Distribute and simplify to get: \( x + 2 = 17 \).

Next, you solve the simplified equation. To isolate \( x \), subtract 2 from both sides:

• This leads to: \( x = 15 \).

Once solved, you can use the obtained value of \( x \) to find the consecutive numbers. This process involves translating a word problem into an equation, simplifying it, and using basic algebraic operations to find the solution.

Problem Solving Steps

Effective problem-solving in algebra involves a step-by-step approach. This clear structured method ensures you correctly interpret and solve problems. Here's a breakdown using the example:

• Define the variables: Start with assigning variables to represent the unknowns, e.g., \( x \) for the smaller integer.
• Set up the equation: Use the problem's conditions to form an equation, for instance, translating 'two times the larger minus the smaller equals 17' into \( 2(x + 1) - x = 17 \).
• Simplify the equation: Make the equation easier to solve by combining like terms, which leads to \( x + 2 = 17 \).
• Solve for the unknown: Use algebraic operations (addition, subtraction) to find \( x \), here resulting in \( x = 15 \).
• Verify and interpret: Check your work and interpret results. Confirm the consecutive integer pair: \( 15 \) and \( 16 \).

By following these steps, you ensure that you approach problems methodically and reduce the chance of errors, leading to accurate answers.