Problem 23 - Fundamentals of Mathematics

Question

Three consecutive even integers add to -36 . What are they?

Step-by-Step Solution

Answer

The integers are -14, -12, and -10.

Step 1: Define Variables

Let's define the three consecutive even integers. If we start with the first integer as \( x \), then the second integer will be \( x + 2 \) and the third will be \( x + 4 \). This setup covers three consecutive even integers.

Step 2: Set Up the Equation

Given that the sum of these three integers is -36, we can write the equation: \[ x + (x + 2) + (x + 4) = -36. \] This equation represents the sum of the three consecutive even integers.

Step 3: Simplify the Equation

Combine like terms in the equation: \[ x + x + 2 + x + 4 = -36 \] simplifies to \[ 3x + 6 = -36. \]

Step 4: Solve for x

Subtract 6 from both sides to start isolating \( x \): \[ 3x + 6 - 6 = -36 - 6 \] simplifies to \[ 3x = -42. \] Now, divide both sides by 3 to solve for \( x \): \[ x = -14. \]

Step 5: Find the Integers

Now that we have \( x = -14 \), calculate the three integers. The first integer is \( x = -14 \), the second is \( x + 2 = -12 \), and the third is \( x + 4 = -10 \). Therefore, the integers are -14, -12, and -10.

Key Concepts

even integers

Even integers are whole numbers that are divisible by 2 without leaving a remainder. You know these numbers as 0, 2, 4, -2, -4, etc. They alternate with odd integers, creating a steady pattern: even, odd, even, odd, and so forth.

When dealing with problems involving even integers, particularly consecutive ones, it's helpful to remember their pattern. If you have one even integer like -14, the next one will be -12, followed by -10, and so on. This is because each consecutive even integer differs from the last one by 2.

To express consecutive even integers algebraically, we start with one even integer, often called \( x \). The next consecutive even integer is \( x + 2 \), and the one after that is \( x + 4 \). This notation helps to simplify calculations and understand the relationships between the numbers.

equations

Equations are mathematical statements expressing the equality between two expressions. They often involve variables and constants. In this problem, we used an equation to find three consecutive even integers that add up to -36. This is typically done by writing the sum as an equation, which is expressed as:

• First Integer: \( x \)
• Second Integer: \( x + 2 \)
• Third Integer: \( x + 4 \)
• Equation: \( x + (x + 2) + (x + 4) = -36 \)

By combining like terms, the equation becomes \( 3x + 6 = -36 \).

The equality tells us that when these numbers are summed, they equal -36. Solving the equation means finding the value of \( x \), which makes the equation true. This requires manipulating the equation using basic arithmetic operations to isolate the variable on one side.

integer arithmetic

Integer arithmetic refers to operations involving whole numbers, such as addition, subtraction, multiplication, and division. In this exercise, integer arithmetic helps solve for the consecutive even numbers by rearranging and simplifying the given equation.

To solve \( 3x + 6 = -36 \):

• We first subtract 6 from each side, eliminating the constant from the left, simplifying it to \( 3x = -42 \).
• Next, divide both sides by 3 to isolate \( x \), yielding \( x = -14 \).

The operations used here demonstrate rearranging equations through basic integer arithmetic. These steps are crucial for isolating variables and finding their values. This approach can be applied to various equations, illustrating the importance of integer arithmetic in algebra. Solving equations involves consistent and logical application of these arithmetic steps.