Problem 35 - Fundamentals of Mathematics

Question

Find three consecutive even integers that add to -24 .

Step-by-Step Solution

Answer

The three consecutive even integers are -10, -8, and -6.

Step 1: Understand the problem

We need to find three consecutive even integers whose sum is -24. Even integers are numbers that are divisible by 2.

Step 2: Define a variable

Let's let the first even integer be represented by the variable \( x \). The next consecutive even integers can then be represented as \( x + 2 \) and \( x + 4 \) respectively.

Step 3: Set up the equation

Since the sum of these three integers is -24, we can set up the equation: \( x + (x + 2) + (x + 4) = -24 \).

Step 4: Simplify the equation

Combine like terms in the equation: \( 3x + 6 = -24 \).

Step 5: Solve for x

First, subtract 6 from both sides to get \( 3x = -30 \). Then, divide by 3 to find \( x = -10 \).

Step 6: Find the consecutive integers

Now that we know \( x = -10 \), calculate the integers: first integer is -10, second is \( -10 + 2 = -8 \), and third is \( -10 + 4 = -6 \).

Step 7: Verify the solution

Check the sum of the three integers: \(-10 + (-8) + (-6) = -24\). The solution satisfies the original condition.

Key Concepts

Even Integers

Even integers are numbers that can be evenly divided by 2 without leaving a remainder. Some examples include numbers such as 2, 4, 6, -2, -4, -6, and so forth. They appear consistently on the number line. Every second number you'll count would be an even integer. This feature makes them predictable, as you can always obtain the next even integer by simply adding 2.

• Even integers include both positive and negative numbers.
• They are positioned equally spaced on both sides of zero.
• Quintessentially, even numbers follow the structure of 2n, where n is any integer.

When we explore problems involving consecutive even integers, identifying this sequence is crucial for setting up the problem correctly. For example, if you start at -10, the next few consecutive even numbers are -8, -6, and so on.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. It often requires a series of logical steps where you manipulate the equation to isolate the variable. This begins with writing the equation based on the information given in a problem.
Here, we set up an equation using variables to find consecutive integers:

• Start with a variable, usually denoted by a letter like x. Here, we let the first integer be x.
• Construct an expression for the sequence, such as x+2 and x+4 for the next two consecutive even integers.
• Formulate the equation based on these expressions to represent the problem situation, like ensuring their sum equals the given total (-24).

Next, you simplify and solve the equation:

• Combine like terms to make the equation easier to solve. In this case, combine x, x+2, and x+4 to simplify to 3x + 6.
• Manipulate the equation to find x, initially by subtracting 6 from both sides, then dividing by 3 to solve for the variable.

Ultimately, solving equations is about finding the balance where one side equals the other.

Integer Sum Problems

Integer sum problems focus on finding integers that meet a certain sum requirement. These problems are common in math exercises and are perfect for practicing conceptual skills with algebra and equations.
When tackling integer sum problems:

• Define each integer uniquely using algebra, often through variables, like we used x for the first integer.
• Express the sum of the integers with an equation, which should equal the specified sum.
• Work systematically through the equation by simplifying and isolating variables until you find the posing integers.

Checking work is crucial for these types of problems. Verification involves substituting the integers back into the original equation to ensure they indeed sum to the given total. This step, as in the solved problem above, reassures accuracy in calculations and reinforces the overarching math concept. Integer sum problems like these build confidence in mathematical reasoning.