Solving Linear Equations Step-by-Step Solve 5 - 2a = -13
Introduction
In the realm of algebra, solving equations is a fundamental skill. This article provides a detailed walkthrough to solve the linear equation 5 - 2a = -13. We will break down each step, ensuring clarity and understanding for anyone looking to master this essential mathematical technique. Linear equations, at their core, are algebraic equations where the highest power of the variable is 1. These equations can be visually represented as a straight line on a graph, hence the name "linear." Solving them involves isolating the variable, in this case, 'a,' to find its value that satisfies the equation. This process often requires the application of various algebraic principles, such as the properties of equality, which allow us to manipulate equations while maintaining their balance. Mastering the art of solving linear equations is not just about finding the correct answer; it's about developing a systematic approach to problem-solving, a skill that extends far beyond the classroom. Whether you're a student grappling with algebra for the first time or someone looking to brush up on your mathematical abilities, understanding linear equations is a crucial step. So, let's dive into the world of algebra and tackle the equation 5 - 2a = -13 together, step by step, unraveling the mystery and discovering the value of 'a' that makes the equation true.
Understanding the Equation: 5 - 2a = -13
Before diving into the solution, let's break down the equation 5 - 2a = -13. This equation states that if we subtract twice the value of 'a' from 5, the result will be -13. Our goal is to find the specific value of 'a' that makes this statement true. Understanding the components of the equation is crucial for a successful solution. The equation comprises constants (numbers), variables (represented by letters), and operations (addition, subtraction, multiplication, division). In this case, 5 and -13 are constants, 'a' is the variable we aim to solve for, and the operations involved are subtraction and multiplication. The term '-2a' signifies -2 multiplied by 'a.' The equal sign (=) signifies that the expressions on both sides of the equation are equivalent. Therefore, whatever we do to one side of the equation, we must also do to the other side to maintain this equivalence. This principle is the cornerstone of solving equations. By understanding the structure and components of the equation, we set the stage for a systematic approach to finding the value of 'a.' Recognizing the relationships between the constants, variables, and operations allows us to strategically manipulate the equation, isolating 'a' and ultimately uncovering its value. The ability to deconstruct equations like this is a fundamental skill in algebra and forms the basis for solving more complex mathematical problems.
Step 1: Isolating the Term with 'a'
The first step in solving 5 - 2a = -13 is to isolate the term containing 'a', which is '-2a'. To do this, we need to eliminate the constant term '5' from the left side of the equation. This is achieved by performing the inverse operation. Since 5 is being added (or, more accurately, a positive 5 is present), we subtract 5 from both sides of the equation. This ensures that the equation remains balanced, adhering to the fundamental principle that any operation performed on one side must be mirrored on the other. Subtracting 5 from both sides yields the following transformation: 5 - 2a - 5 = -13 - 5. Simplifying this, we get -2a = -18. Now, the term with 'a' is isolated on the left side, making the next step of solving for 'a' more straightforward. This process of isolating the variable term is a cornerstone of solving equations. It involves strategically applying inverse operations to eliminate unwanted terms, paving the way for the variable to be the sole focus. By meticulously removing the constants surrounding the variable term, we gradually unravel the equation, bringing us closer to the solution. This initial isolation step is not just a mathematical maneuver; it's a crucial strategic move that sets the stage for the subsequent steps in the equation-solving process. It's about creating a clear path towards the ultimate goal: discovering the value of 'a'.
Step 2: Solving for 'a'
Now that we have -2a = -18, the next step is to isolate 'a' completely. Currently, 'a' is being multiplied by -2. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by -2, again maintaining the balance of the equation. This gives us (-2a) / -2 = (-18) / -2. On the left side, -2 divided by -2 equals 1, leaving us with 'a'. On the right side, -18 divided by -2 equals 9. Therefore, we arrive at the solution: a = 9. This final step is where the value of the variable is revealed. The division process effectively cancels out the coefficient of 'a,' leaving 'a' standing alone and exposed its true value. This is the culmination of the step-by-step process, where all the previous manipulations have led to this singular moment of discovery. Solving for 'a' is not just about performing a division; it's about completing the algebraic puzzle. It's the moment where the unknown becomes known, where the variable transforms into a specific value. The satisfaction of arriving at this solution underscores the power of algebraic manipulation and the systematic approach to problem-solving that we've employed. This step solidifies our understanding of the equation and confirms the value of 'a' that makes the equation true.
Verification: Checking the Solution
To ensure the accuracy of our solution, it's crucial to verify it. We substitute a = 9 back into the original equation: 5 - 2a = -13. Replacing 'a' with 9, we get 5 - 2(9) = -13. Now, we simplify the left side of the equation. 2 multiplied by 9 is 18, so we have 5 - 18 = -13. Subtracting 18 from 5 gives us -13. Thus, the equation becomes -13 = -13, which is a true statement. This confirms that our solution, a = 9, is correct. Verification is a vital step in the equation-solving process. It acts as a safety net, catching any potential errors that might have occurred during the manipulation of the equation. By substituting the calculated value back into the original equation, we essentially put our solution to the test. If the equation holds true, it provides strong evidence that our solution is accurate. This process not only validates our answer but also reinforces our understanding of the equation and the steps taken to solve it. Verification is not just a formality; it's an integral part of the problem-solving process, ensuring confidence in our results and fostering a deeper comprehension of algebraic principles. It's the final checkmark that confirms our journey to the solution has been successful.
Conclusion
In this article, we have successfully solved the linear equation 5 - 2a = -13, finding that a = 9. We achieved this by systematically isolating the variable 'a' through the application of inverse operations and verifying our solution to ensure accuracy. This step-by-step approach provides a clear framework for solving similar equations. Mastering the art of solving linear equations is a fundamental skill in algebra and a valuable tool for problem-solving in various contexts. The ability to manipulate equations, isolate variables, and verify solutions is a testament to mathematical proficiency and a key component of logical thinking. The journey through this equation has not only yielded a specific answer but also illuminated the broader principles of algebraic manipulation. The strategic use of inverse operations, the importance of maintaining balance in an equation, and the critical role of verification are all lessons that extend beyond this particular problem. This understanding empowers us to tackle a wide range of mathematical challenges with confidence and precision. The solution to 5 - 2a = -13 is not just a number; it's a symbol of our ability to dissect, analyze, and solve problems systematically, a skill that transcends the boundaries of mathematics and enriches our approach to the world around us.
