Step-by-Step Algebraic Equation Solving

by Alex Johnson 40 views

Are you looking to master the art of solving algebraic equations? You've come to the right place! In this article, we'll break down the process of solving equations, using Marlena's excellent work as our guide. Marlena tackled the equation 2x+5=−10−x2x + 5 = -10 - x, and her methodical approach is a fantastic way to learn. We'll explore each step, explaining the underlying mathematical principles and why each move is crucial for arriving at the correct solution. Understanding these steps isn't just about getting the right answer; it's about building a strong foundation in algebra, which is essential for more complex mathematical concepts down the line. Whether you're a student learning algebra for the first time or someone looking for a refresher, this guide will provide clear explanations and helpful insights. We'll delve into the properties of equality, the importance of isolating variables, and how to check your work. So, let's dive in and demystify algebraic equations together!

Understanding the Goal: Isolating the Variable

The primary goal when solving linear equations like the one Marlena worked on, 2x+5=−10−x2x + 5 = -10 - x, is to isolate the variable. This means getting the variable (in this case, 'xx') all by itself on one side of the equals sign. Think of it like a balancing act. Whatever you do to one side of the equation, you must do to the other side to keep the equation balanced and true. This principle is known as the Addition Property of Equality and the Multiplication Property of Equality. Marlena's journey to finding 'x=−5x = -5' is a perfect illustration of applying these properties. She started with an equation where 'xx' appeared on both sides, and her steps systematically moved terms around until 'xx' was isolated. We'll go through each of her steps, explaining the specific operation used and its purpose in moving us closer to the solution. This process is fundamental not only in algebra but also in many real-world applications, from calculating distances to managing finances. By understanding how to manipulate equations, you gain a powerful tool for problem-solving and logical reasoning. It’s about developing a systematic approach to breaking down complex problems into manageable steps, a skill valuable in any field.

Step 1: Combining Like Terms and Isolating the Variable Term

Marlena's first step was to transform 2x+5=−10−x2x + 5 = -10 - x into 3x+5=−103x + 5 = -10. Let's break down how she achieved this. The equation initially has 'xx' terms on both sides: '2x2x' on the left and '−x-x' on the right. To begin isolating the 'xx' terms on one side, Marlena chose to add 'xx' to both sides of the equation. This is a direct application of the Addition Property of Equality. Adding 'xx' to both sides maintains the equality of the equation. On the left side, '2x+x2x + x' combines to become '3x3x'. On the right side, '−x+x-x + x' cancels out, leaving just '−10-10'. The '+5+ 5' on the left side remains untouched in this step. So, after adding 'xx' to both sides, the equation transforms into 3x+5=−103x + 5 = -10. This is a crucial step because it consolidates all the 'xx' terms into a single term, making it easier to manage. It also moves us closer to the goal of isolating the 'xx' term itself, as we now have '3x3x' on one side and a constant term on the other. This strategic move sets the stage for the next simplification. The key here is understanding that adding the same quantity to both sides doesn't change the solution set of the equation. It’s a fundamental rule that underpins all algebraic manipulation. The goal is to simplify the equation progressively, removing complexities until the unknown variable stands alone. Marlena's execution of this step is clean and efficient, demonstrating a good grasp of algebraic manipulation. It's about preparing the equation for the final isolation of 'xx'.

Step 2: Isolating the Variable Term Further

Following the first step, Marlena had the equation 3x+5=−103x + 5 = -10. Her next move was to get the '3x3x' term by itself on the left side. To do this, she needed to eliminate the '+5+ 5' from the left side. She accomplished this by subtracting '55' from both sides of the equation. This again uses the Addition Property of Equality (or, more specifically, the Subtraction Property of Equality, which is a consequence of adding a negative number). Subtracting '55' from the left side cancels out the '+5+ 5' (since 5−5=05 - 5 = 0), leaving just '3x3x'. On the right side, she subtracted '55' from '−10-10', resulting in '−15-15'. So, the equation becomes 3x=−153x = -15. This step is vital because it isolates the term containing the variable. We've moved from an equation with a variable term and a constant term on one side (3x+53x + 5) to an equation where only the variable term (3x3x) remains on that side. Each operation is performed with the intention of moving closer to the goal of having 'xx' alone. This systematic approach ensures that we don't lose track of the equality. The concept of inverse operations is at play here: addition and subtraction are inverse operations. Since '55' was added to '3x3x', we use subtraction to undo that addition. The principle of performing the same operation on both sides is paramount to maintaining the equation's balance. Marlena's efficiency in this step further streamlines the process, bringing her very close to the final solution. It demonstrates a clear understanding of how to manipulate equations to isolate the part that contains the unknown.

Step 3: Solving for the Variable

Marlena's equation at this point is 3x=−153x = -15. The final step to solve for 'xx' involves undoing the multiplication. The term '3x3x' means '33 multiplied by xx'. To isolate 'xx', Marlena must perform the inverse operation of multiplication, which is division. She divided both sides of the equation by '33'. This is an application of the Multiplication Property of Equality (specifically, dividing by a non-zero number is equivalent to multiplying by its reciprocal). Dividing the left side by '33' leaves 'xx' (since 3x/3=x3x / 3 = x). On the right side, she divided '−15-15' by '33', which equals '−5-5'. Thus, the equation simplifies to x=−5x = -5. This is the solution to the original equation. This final step is where the variable is fully isolated, giving us its numerical value. It's the culmination of all the previous steps, each designed to simplify the equation and move the variable into a position where it could be isolated. The consistent application of inverse operations and the principle of maintaining balance by performing the same action on both sides are what make this process reliable. Marlena's correct execution of this division leads directly to the final answer, demonstrating a complete understanding of how to solve this type of linear equation. Checking the answer is also a good practice, which we'll touch upon next.

Checking Your Work: Ensuring Accuracy

After arriving at a solution, like Marlena's x=−5x = -5, it's always a good practice to check your work. This involves substituting the value you found back into the original equation to see if it makes the equation true. Let's do that for Marlena's solution. The original equation was 2x+5=−10−x2x + 5 = -10 - x. Substituting x=−5x = -5:

Left side: 2(−5)+5=−10+5=−52(-5) + 5 = -10 + 5 = -5

Right side: −10−(−5)=−10+5=−5-10 - (-5) = -10 + 5 = -5

Since the left side (−5-5) equals the right side (−5-5), the equation holds true. This confirms that Marlena's solution x=−5x = -5 is indeed correct. Checking your solution is a vital part of the problem-solving process in mathematics. It builds confidence in your answer and helps catch any potential errors made during the calculation. It reinforces the understanding that an equation is a statement of equality, and the goal is to find the value(s) that make that statement true. This verification step is just as important as the steps taken to find the solution itself, as it guarantees the accuracy of your findings. It’s a small extra effort that can save a lot of trouble by ensuring you haven’t made a slip-up along the way. For anyone learning algebra, making this a habit will significantly improve their accuracy and understanding.

Conclusion: The Power of Systematic Algebra

Marlena's methodical approach to solving the equation 2x+5=−10−x2x + 5 = -10 - x highlights the power of systematic algebraic manipulation. Each step she took was logical and justified by fundamental properties of equality: the addition property and the multiplication property. By consistently applying these principles – adding or subtracting the same value from both sides to isolate terms, and multiplying or dividing both sides by the same non-zero value to isolate the variable – she was able to transform a complex-looking equation into a simple statement of the variable's value. The process involved combining like terms, isolating the variable term, and finally, solving for the variable. Remembering to check your answer by substituting it back into the original equation provides a crucial layer of verification, ensuring accuracy. Mastering these techniques is fundamental to success in mathematics, opening doors to more advanced topics and real-world applications where problem-solving through equations is essential. Keep practicing these steps, and you'll find that solving algebraic equations becomes increasingly intuitive and manageable.

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