Five Consecutive Even Integers Sum To -70 Solution And Explanation

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In the realm of mathematics, solving problems involving consecutive integers is a fundamental concept often encountered in algebra. This article delves into a specific problem: finding five consecutive even integers that sum up to -70. We'll explore the steps involved in setting up the equation, solving for the unknown, and ultimately identifying the five integers. Understanding how to tackle such problems not only enhances your algebraic skills but also provides a framework for approaching similar mathematical challenges. Let's embark on this journey of mathematical exploration and unravel the solution together.

Understanding Consecutive Even Integers

When dealing with consecutive even integers, it's crucial to grasp the pattern they follow. Even integers are whole numbers divisible by 2 (e.g., -4, -2, 0, 2, 4). Consecutive even integers are even numbers that follow each other in sequence, with a difference of 2 between each number. For example, -6, -4, -2, 0, and 2 are consecutive even integers. To represent these integers algebraically, we can use a variable, typically 'x', to denote the first even integer in the sequence. The next consecutive even integer would then be 'x + 2', followed by 'x + 4', 'x + 6', and so on. This algebraic representation allows us to formulate equations and solve problems involving consecutive even integers. In our specific problem, we are tasked with finding five such integers that meet a particular condition: their sum must equal -70. By understanding the nature of consecutive even integers and their algebraic representation, we can effectively approach this problem and determine the solution. Let's proceed to the next step, where we will set up the equation to solve for these integers.

Setting Up the Equation

The key to solving this problem lies in translating the word problem into a mathematical equation. We know we need to find five consecutive even integers that add up to -70. As we discussed earlier, we can represent these integers algebraically. Let's denote the first even integer as 'x'. Then, the next four consecutive even integers would be 'x + 2', 'x + 4', 'x + 6', and 'x + 8'. The problem states that the sum of these five integers is -70. Therefore, we can write the equation as follows:

x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = -70

This equation forms the foundation for our solution. It encapsulates the relationship between the five consecutive even integers and their sum. The left side of the equation represents the sum of the five integers, while the right side represents the target sum, which is -70. Now that we have successfully set up the equation, the next step is to simplify and solve it for the unknown variable 'x'. This will give us the value of the first even integer, which we can then use to find the remaining four integers. Let's proceed to the next section where we will delve into the process of solving this equation.

Solving the Equation

Now that we have the equation:

x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = -70

The first step in solving this equation is to simplify it by combining like terms. We have five 'x' terms, which can be combined as 5x. The constant terms (2, 4, 6, and 8) can be added together: 2 + 4 + 6 + 8 = 20. So, the simplified equation becomes:

5x + 20 = -70

Next, we want to isolate the term with the variable 'x'. To do this, we subtract 20 from both sides of the equation:

5x + 20 - 20 = -70 - 20

This simplifies to:

5x = -90

Finally, to solve for 'x', we divide both sides of the equation by 5:

5x / 5 = -90 / 5

This gives us:

x = -18

So, the first even integer in our sequence is -18. Now that we have the value of 'x', we can find the remaining four consecutive even integers. Let's move on to the next section to determine the complete set of integers.

Identifying the Integers

We've successfully solved for 'x' and found that the first even integer is -18. Now, to find the remaining four consecutive even integers, we simply add 2 to the previous integer. So, the sequence is as follows:

  • First integer: x = -18
  • Second integer: x + 2 = -18 + 2 = -16
  • Third integer: x + 4 = -18 + 4 = -14
  • Fourth integer: x + 6 = -18 + 6 = -12
  • Fifth integer: x + 8 = -18 + 8 = -10

Therefore, the five consecutive even integers that sum up to -70 are -18, -16, -14, -12, and -10. To verify our solution, we can add these integers together:

-18 + (-16) + (-14) + (-12) + (-10) = -70

The sum indeed equals -70, confirming that our solution is correct. We have successfully identified the five consecutive even integers that satisfy the given condition. This problem demonstrates the power of algebraic representation and equation-solving in tackling mathematical challenges. In the next section, we'll summarize the steps we took to solve this problem and highlight the key concepts involved.

Conclusion

In this article, we tackled the problem of finding five consecutive even integers that sum up to -70. We began by understanding the nature of consecutive even integers and how to represent them algebraically. We then translated the word problem into a mathematical equation, which was the crucial step in solving the problem. The equation was:

x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = -70

We simplified the equation by combining like terms and then isolated the variable 'x' to solve for its value. We found that x = -18, which represents the first even integer in the sequence. Using this value, we determined the remaining four integers by adding 2 consecutively. The five integers were found to be -18, -16, -14, -12, and -10. We verified our solution by adding these integers together, confirming that their sum is indeed -70. This problem highlights the importance of algebraic representation, equation-solving, and logical reasoning in mathematics. By understanding these concepts, we can approach and solve a wide range of mathematical problems. The ability to translate word problems into equations and then solve those equations is a fundamental skill in algebra and mathematics in general. This article has provided a step-by-step guide to solving such problems, and the principles discussed can be applied to similar challenges involving consecutive integers or other sequences. Remember, practice is key to mastering these skills, so continue to explore and solve different types of problems to enhance your mathematical abilities.