Problem 25 - Elementary Algebra

Question

Set up an algebraic equation and then solve. If the smaller of two consecutive even integers is subtracted from 3 times the larger the result is 42. Find the integers.

Step-by-Step Solution

Answer

The integers are 18 and 20.

Step 1: Define Variables

Let the smaller even integer be \( x \). Since they are consecutive even integers, the larger one is \( x + 2 \).

Step 2: Translate the Problem into an Equation

We need to find \( 3 \times \) the larger integer minus the smaller integer gives 42. That translates to the equation: \[ 3(x + 2) - x = 42 \]

Step 3: Simplify the Equation

Simplify the left side of the equation from the previous step: \[ 3(x + 2) - x = 3x + 6 - x = 2x + 6 \]

Step 4: Solve for x

Set the equation \( 2x + 6 = 42 \). Subtract 6 from both sides: \[ 2x = 36 \]Next, divide by 2: \[ x = 18 \]

Step 5: Find the Larger Integer

Since \( x = 18 \), the larger consecutive even integer is \( x + 2 = 20 \).

Step 6: Verify the Solution

Verify that \( 3 \times 20 - 18 = 60 - 18 = 42 \), which matches the problem statement.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow one another in order, each being one unit or a fixed value more than the previous one. In this specific problem, we deal with consecutive even integers.
Even integers are numbers divisible by two, typically ending in 0, 2, 4, 6, or 8. Consecutive even integers are numbers that are two units apart from each other.
For example, if we have 4 as one even integer, the next consecutive even integer would be 6.

• This means, if "x" is our smaller even integer, the next consecutive even integer is "x + 2".
• This difference of 2 units is crucial for solving algebraic problems involving consecutive even integers.

Understanding this concept allows us to correctly set up equations for problems that require finding consecutive numbers.

Variable Representation

Variable representation is a fundamental concept in algebra, where unknown values are represented by symbols, usually letters like \( x \) or \( y \). This helps in forming equations and solving problems with unknowns.
In this exercise, we represent our smaller even integer as \( x \).
Since we are dealing with consecutive even integers, the next consecutive integer becomes \( x + 2 \).
By assigning these variables, it becomes easier to manipulate and solve the given problem through algebraic equations.

• Using variables streamlines the process of expression and enables us to work with unknowns in mathematical statements.
• The approach of defining even integers in terms of \( x \) and \( x + 2 \) sets up a straightforward path to translate the word problem into a solvable equation.

This method transforms a verbal situation into a mathematical one.

Equation Solving

Equation solving is the process of finding the value of unknown variables that satisfy the given equation.
In this problem, we translated the word statement into the equation:
\[ 3(x + 2) - x = 42 \]
The goal is to simplify and solve this equation step-by-step. Here’s how:

• Simplify the equation by distributing 3 into \( x + 2 \): \[ 3x + 6 \]
• Subtract \( x \) from \( 3x + 6 \), which simplifies to \( 2x + 6 \)
• To isolate \( x \), subtract 6 from both sides: \( 2x = 36 \)
• Finally, divide by 2 to solve for \( x \): \( x = 18 \)

This process of simplifying and solving transforms our word problem into a clear solution, revealing \( x = 18 \) as the value of the smaller integer.

Verification of Solutions

Verification of solutions is an essential step in math to ensure that the solution is correct and satisfies the original problem conditions. This gives us confidence in the accuracy of our answer.
To verify, we substitute our solution back into the original situation described:

• We found \( x = 18 \), making the larger integer \( x + 2 = 20 \).
• Calculate \( 3 \times 20 - 18 \).
• That equals \( 60 - 18 = 42 \), which matches the initial problem condition.

By recreating the problem situation with our solution, we confirm its validity.
This practice of checking reinforces our confidence and ensures accuracy in algebraic problem-solving.