Problem 75 - Elementary Algebra

Question

A larger integer is 3 more than twice a smaller integer. If their sum is 39 , then find the integers.

Step-by-Step Solution

Answer

The integers are 12 and 27.

Step 1: Define Variables

Let's denote the smaller integer by \( x \). According to the problem, the larger integer is 3 more than twice the smaller integer. Therefore, we can write the larger integer as \( 2x + 3 \).

Step 2: Write the Equation

Based on the information that their sum is 39, we can write the equation: \( x + (2x + 3) = 39 \).

Step 3: Simplify the Equation

Simplify the equation by combining like terms: \( x + 2x + 3 = 39 \) simplifies to \( 3x + 3 = 39 \).

Step 4: Solve for the Smaller Integer

Subtract 3 from both sides of the equation: \( 3x + 3 - 3 = 39 - 3 \), which simplifies to \( 3x = 36 \). Now, divide both sides by 3: \( \frac{3x}{3} = \frac{36}{3} \), resulting in \( x = 12 \).

Step 5: Solve for the Larger Integer

Substitute \( x = 12 \) back into the expression for the larger integer: \( 2x + 3 = 2 \times 12 + 3 = 24 + 3 = 27 \).

Step 6: Verify the Solution

Check that the sum of the integers is correct: \( 12 + 27 = 39 \), which matches the problem statement.

Key Concepts

Understanding Integer Variables

When solving linear equations involving integers, it's crucial to define integer variables clearly. In our exercise, we are dealing with two integers: a smaller one and a larger one. To simplify and solve the problem, we use a variable, often symbolized as \(x\), to represent one of these unknown integers.
Why integers, you might ask? Integers are whole numbers that can be positive, negative, or zero. In many mathematical problems, using integer variables is necessary when you want to find whole number solutions.
It helps to:

• Represent quantities easily in equations.
• Simplify the process of solving qualitative problems.
• Ensure availability of exact and clean results, as opposed to fractions or decimals.

Defining your variables appropriately is the first step towards solving any problem efficiently.

Solving Linear Equations

Equation solving is the process by which we find the value(s) of the variables that satisfy an equation. In our original exercise, once we have set up the variables, the next step is to construct the equation based on the given relationships.
The exercise describes a situation where one integer is 3 more than twice another and gives a total sum of 39. This enables us to set up an equation: \(x + (2x + 3) = 39\).
Here's a straightforward method to solve linear equations:

• First, simplify by combining like terms. In this case, \(x + 2x + 3\) becomes \(3x + 3\).
• Then, isolate the term with the variable by removing any constants on the side of the equation with the variable (e.g., subtract 3 from both sides).
• Finally, divide to solve for the variable (e.g., divide both sides by 3 to find \(x = 12\)).

This systematic approach will guide you to find values of the unknowns confidently.

Verifying Your Solution

Verification is a vital final step in solving equations. This process ensures that the solution obtained is correct and satisfies the original problem conditions.
Why verify? Because a simple arithmetic error or misunderstanding can lead to incorrect results. By substituting back into the original equation or using logical reasoning, you confirm your solutions.
In our exercise, once we found \(x = 12\) and the larger number was calculated as \(2x + 3 = 27\), we checked:

• Ensure that adding the integers gives the total sum: \(12 + 27 = 39\).

Since this matches the given condition, we concluded that the integers found are indeed the correct solution. Verification not only adds a layer of confidence to your answer but it also strengthens problem-solving skills overall.