Solving For X In The Equation [(-23) + 15] + (-33) = (-23) + [15 + X]
Introduction
In this article, we will delve into solving a fundamental algebraic equation. The equation we aim to solve is: [(-23) + 15] + (-33) = (-23) + [15 + x]. This equation involves integer addition and the application of the associative property. Our primary goal is to find the value of x that satisfies this equation. Understanding how to solve such equations is crucial for mastering basic algebra and building a strong foundation for more complex mathematical concepts. This exercise will not only enhance your problem-solving skills but also give you a clear understanding of how mathematical operations work with negative numbers and variables. The step-by-step approach we will use ensures clarity and makes the solution easy to follow, even for those who are relatively new to algebra.
Understanding the Equation
Before we dive into solving the equation, let's first understand its structure. The equation [(-23) + 15] + (-33) = (-23) + [15 + x] demonstrates the associative property of addition, which states that the way in which numbers are grouped in an addition operation does not change the sum. This property is a cornerstone of arithmetic and algebra, allowing us to manipulate equations to isolate variables and find solutions. In this particular equation, we have integers and a variable, x. The left side of the equation involves adding -23 and 15, and then adding -33 to the result. The right side of the equation involves adding 15 and x, and then adding -23 to the result. To solve for x, we need to simplify both sides of the equation and then isolate x on one side. This involves performing arithmetic operations and using algebraic manipulation techniques. Understanding the properties of numbers and operations is essential in solving such equations effectively.
Step-by-Step Solution
To solve the equation [(-23) + 15] + (-33) = (-23) + [15 + x], we will follow a step-by-step approach to ensure clarity and accuracy. Each step will involve simplifying the equation and moving closer to isolating the variable x. This methodical approach is crucial for avoiding errors and understanding the underlying mathematical principles. Let's begin by simplifying both sides of the equation.
Step 1: Simplify the Left Side of the Equation
First, let's simplify the left side of the equation: [(-23) + 15] + (-33). We begin by adding -23 and 15. When adding a negative number and a positive number, we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. In this case, |-23| = 23 and |15| = 15. So, 23 - 15 = 8. Since -23 has a larger absolute value, the result is -8. Now we have (-8) + (-33). Adding two negative numbers involves adding their absolute values and keeping the negative sign. Thus, 8 + 33 = 41, and the result is -41. So, the simplified left side of the equation is -41. This step demonstrates the basic rules of integer addition and the importance of handling signs correctly.
Step 2: Simplify the Right Side of the Equation
Next, we simplify the right side of the equation: (-23) + [15 + x]. Here, we have a variable x that we need to isolate. We can start by simplifying the expression inside the brackets. However, since x is an unknown variable, we cannot perform a numerical addition with 15. So, we leave the expression as 15 + x. Now, the right side of the equation becomes (-23) + (15 + x). To further simplify, we can add -23 to the expression (15 + x). This means we are adding -23 and 15, which we already calculated in Step 1 to be -8. So, we have -8 + x. Therefore, the simplified right side of the equation is -8 + x. This step highlights how algebraic expressions are handled and how variables are kept intact during simplification.
Step 3: Set the Simplified Sides Equal
Now that we have simplified both sides of the equation, we can set them equal to each other. From Step 1, the left side simplifies to -41. From Step 2, the right side simplifies to -8 + x. So, our equation now looks like this: -41 = -8 + x. This step is crucial because it brings together the simplified expressions, allowing us to focus on isolating x.
Step 4: Isolate x
To isolate x, we need to get x by itself on one side of the equation. We can do this by adding 8 to both sides of the equation. This will cancel out the -8 on the right side. So, we add 8 to both sides: -41 + 8 = -8 + x + 8. On the left side, -41 + 8 is calculated by subtracting 8 from 41 and keeping the negative sign, which gives us -33. On the right side, -8 + 8 cancels out, leaving us with just x. So, the equation becomes -33 = x. This step demonstrates the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain balance and isolate the variable.
Step 5: Final Solution
We have now isolated x and found its value. From Step 4, we have -33 = x. Therefore, the value of x that satisfies the original equation is -33. This is our final solution. We have successfully solved the equation by simplifying both sides, setting them equal, and isolating the variable. This step concludes our solution process and provides the answer to the problem.
Verification
To ensure our solution is correct, we can substitute x = -33 back into the original equation and check if both sides of the equation are equal. The original equation is [(-23) + 15] + (-33) = (-23) + [15 + x]. Substituting x = -33, we get [(-23) + 15] + (-33) = (-23) + [15 + (-33)]. First, simplify the left side: [(-23) + 15] + (-33) = (-8) + (-33) = -41. Next, simplify the right side: (-23) + [15 + (-33)] = (-23) + (-18) = -41. Since both sides of the equation equal -41, our solution x = -33 is correct. This verification step is a crucial part of the problem-solving process, as it confirms the accuracy of our calculations and algebraic manipulations. By verifying our solution, we can be confident in our answer and the methods we used to obtain it.
Conclusion
In this article, we successfully solved the equation [(-23) + 15] + (-33) = (-23) + [15 + x] by following a step-by-step approach. We began by understanding the equation and the properties of addition, then simplified both sides of the equation, set them equal, and isolated the variable x. Our final solution was x = -33, which we verified by substituting it back into the original equation. This exercise demonstrates the importance of mastering basic algebraic principles and techniques. By understanding how to manipulate equations and isolate variables, we can solve a wide range of mathematical problems. The methodical approach used here, involving simplification, equation setting, isolation, and verification, is a valuable skill that can be applied to various mathematical challenges. Through practice and careful application of these techniques, anyone can improve their problem-solving abilities and gain a deeper understanding of algebra.
