Solving For X: -2x + 3 = -15
Let's dive into the world of algebra and tackle the equation -2x + 3 = -15. This article provides a step-by-step guide on how to solve for x in this equation. Understanding how to solve linear equations is a fundamental skill in mathematics, and this example provides a great opportunity to hone those skills. We will break down each step, ensuring that you grasp the underlying concepts and can confidently solve similar problems in the future. So, grab your pencil and paper, and let's get started on this algebraic adventure!
Understanding Linear Equations
Before we jump into the solution, let's briefly discuss what a linear equation is. A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable we want to solve for. In our case, the equation -2x + 3 = -15 perfectly fits this form. The goal is to isolate x on one side of the equation to determine its value. This involves performing operations on both sides of the equation to maintain balance and gradually simplify it. Remember, whatever you do to one side, you must do to the other to keep the equation true. This principle is the cornerstone of solving algebraic equations, ensuring that the relationship between the two sides remains consistent throughout the process.
The Importance of Isolating the Variable
The core strategy in solving any equation is isolating the variable. Isolating the variable means getting the variable (in this case, x) by itself on one side of the equation. Once we achieve this, the other side of the equation will reveal the value of the variable. This isolation is achieved by using inverse operations, which are operations that undo each other. For instance, addition and subtraction are inverse operations, as are multiplication and division. By strategically applying these inverse operations, we can gradually strip away the terms surrounding the variable until it stands alone, proudly displaying its solution. This methodical approach ensures accuracy and clarity in the solving process.
Step-by-Step Solution
Now, let's break down the solution to the equation -2x + 3 = -15 step by step:
Step 1: Subtract 3 from both sides
Our first goal is to get rid of the constant term on the left side of the equation. To do this, we subtract 3 from both sides:
-2x + 3 - 3 = -15 - 3
This simplifies to:
-2x = -18
By subtracting 3 from both sides, we've maintained the equation's balance while moving closer to isolating x. This is a critical step, as it eliminates the constant term on the variable's side, making the next step of dividing by the coefficient of x much more straightforward. The principle of maintaining balance by performing the same operation on both sides is fundamental in algebra, ensuring that the equation remains true throughout the solving process. It's like a mathematical seesaw – whatever you do on one side, you must mirror on the other.
Step 2: Divide both sides by -2
Now we have -2x = -18. To isolate x, we need to get rid of the coefficient -2. We do this by dividing both sides of the equation by -2:
-2x / -2 = -18 / -2
This simplifies to:
x = 9
And there you have it! We have successfully isolated x and found its value. Dividing both sides by -2 completes the isolation process, revealing the solution. This step is crucial because it undoes the multiplication that was acting on x. The negative signs cancel out, leading us to a positive value for x. This highlights the importance of paying close attention to signs in algebraic manipulations, as they can significantly impact the final result. The solution x = 9 means that if we substitute 9 for x in the original equation, the equation will hold true.
Verification
To ensure our solution is correct, we can substitute x = 9 back into the original equation:
-2(9) + 3 = -15
Let's simplify:
-18 + 3 = -15
-15 = -15
The equation holds true! This confirms that our solution x = 9 is indeed correct. Verification is a crucial step in the problem-solving process, providing confidence in the accuracy of the solution. By substituting the calculated value back into the original equation, we can ensure that both sides balance out. This step not only confirms the answer but also reinforces the understanding of the equation's meaning. It's like a final checkmark, ensuring that the journey to the solution has been successful.
Conclusion
Therefore, the value of x that makes the equation -2x + 3 = -15 true is 9. By following these step-by-step instructions, you can confidently solve similar linear equations. Remember the key principles: isolating the variable and performing the same operations on both sides of the equation to maintain balance. With practice, these skills will become second nature, and you'll be able to tackle more complex algebraic problems with ease. Solving equations is a fundamental skill in mathematics and has wide-ranging applications in various fields, from science and engineering to finance and economics. So keep practicing and building your algebraic prowess!
For further learning and practice on algebra, you might find helpful resources at Khan Academy Algebra. This external resource can provide additional examples, explanations, and exercises to enhance your understanding and skills in algebra.
