Solving for Three Consecutive Odd Integers Whose Sum is -51
Verifying and solving algebraic equations is a crucial skill in mathematics, often appearing in standardized tests and practical problem-solving scenarios. This article will walk you through the process of determining three consecutive odd integers that sum up to -51. Let's explore the solution step by step.
A Common Approach to Solving Integer Problems
First, let's consider the problem: Find three consecutive odd integers whose sum is -51. A common approach is to express these integers algebraically and then set up an equation based on their sum.
Expressing the Integers Algebraically
We can denote the first odd integer as x. The next two consecutive odd integers can be written as x 2 and x 4. Therefore, we can set up the following equation to represent their sum:
x x 2 x 4 -51Simplifying the Equation
Simplifying this equation, we get:
3x 6 -51Subtracting 6 from both sides, we get:
3x -57Dividing both sides by 3, we find:
x -19Determining the Integers
Now that we have determined x, we can find the three consecutive odd integers:
The first integer is x -19. The second integer is x 2 -19 2 -17. The third integer is x 4 -19 4 -15.Therefore, the three consecutive odd integers that sum up to -51 are -19, -17, and -15.
Alternative Methods and Verification
There might be simpler ways to solve such problems, such as using the average or the middle number. According to one approach, -51 divided by 3 (since we need three numbers that are equally spaced) results in -17, which is the middle number. The previous odd integer is -17 - 2 -19 and the next is -17 2 -15. Hence, the three consecutive odd integers are -19, -17, and -15.
Conclusion
This solution demonstrates how to identify three consecutive odd integers that sum up to a given value. The key is to set up an equation based on the sum and solve for the variable, then use the variable to find the integers. Understanding these steps is essential for solving similar problems in future.