Problem 33 - Fundamentals of Mathematics

Question

Find three consecutive integers that add to -57 .

Step-by-Step Solution

Answer

The three consecutive integers are -20, -19, and -18.

Step 1: Understanding the Problem

We are asked to find three consecutive integers that sum to -57. Consecutive integers follow one after the other without any gap, such as 4, 5, 6.

Step 2: Setting up the Equation

Let's denote the smallest of these three consecutive integers as \( x \). Therefore, the next consecutive integers will be \( x+1 \) and \( x+2 \). The equation for their sum will be: \[ x + (x + 1) + (x + 2) = -57 \]

Step 3: Simplifying the Equation

Combine like terms in the equation: \[ x + x + 1 + x + 2 = -57 \] This simplifies to \[ 3x + 3 = -57 \]

Step 4: Solving for x

We need to isolate \( x \). Start by subtracting 3 from both sides of the equation: \[ 3x + 3 - 3 = -57 - 3 \] This simplifies to \[ 3x = -60 \]. Divide both sides by 3: \[ x = -20 \].

Step 5: Finding the Consecutive Integers

Now that we know \( x = -20 \), the three consecutive integers are \( x = -20 \), \( x+1 = -19 \), and \( x+2 = -18 \).

Key Concepts

Equation Solving

Equation solving is a fundamental skill in mathematics that allows us to find unknown values, called variables. In this exercise, we started by defining our unknown, which was the first of the three consecutive integers, as \( x \).
By expressing the other consecutive integers in terms of \( x \), we set up an equation: \( x + (x + 1) + (x + 2) = -57 \). This step requires us to express real-world situations using mathematical language.

We then simplify the equation by combining like terms, reducing it to \( 3x + 3 = -57 \). Solving this equation involves eliminating constant terms and isolating the variable.

• Subtract 3 from both sides to remove the constant: \( 3x = -60 \).
• Divide by 3 to solve for \( x \): \( x = -20 \).

This process of isolating the variable through algebraic manipulation is a key part of equation solving.

Integer Properties

Integers are whole numbers that include positive, negative numbers, and zero. They have specific properties that make them interesting and unique.

Consecutive integers, like the ones you encountered in this exercise, are integers that follow one another without gaps. If \( x \) is an integer, then \( x + 1 \) and \( x + 2 \) are its consecutive integers. This sequential nature forms the basis for setting up equations when dealing with sequences of integers.

• Consecutive integers differ by 1.
• They follow a simple, predictable pattern, which makes them useful in sequences and series.

Understanding these properties helps us to express relationships between numbers and solve problems that involve multiple sequential values.

Mathematical Problem Solving

Mathematical problem solving involves understanding the problem, setting up a strategy, and systematically solving it to reach a solution.

In this exercise, we began by fully grasping what was being asked: to find three integers that add up to -57. Setting up a clear, concise plan was crucial. By defining \( x \) and expressing its consecutive numbers, we converted the problem from words into an actionable mathematical equation.

This approach emphasizes the importance of:

• Breaking down complex problems into simpler components.
• Using logical reasoning and algebraic manipulation.
• Verifying solutions by checking if they satisfy the original problem statement.

When problem-solving, it's not just about getting the right answer but also about the steps taken to ensure the solution makes sense in the context of the problem.