Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a jumbled mess but secretly holds a simple solution? Well, today, we're diving headfirst into one such equation. We're going to break down the equation 2(x-3)+9=3(x+1)+x step by step, and by the end of this article, you'll be a pro at solving it. So, grab your calculators (or your mental math muscles), and let's get started!
Decoding the Equation: A Step-by-Step Approach
In this section, we're going to meticulously dissect the equation 2(x-3)+9=3(x+1)+x, transforming it from a seemingly complex expression into a clear and manageable form. Our journey begins with understanding the fundamental principles of algebraic manipulation, ensuring that each step we take is grounded in sound mathematical reasoning. Let's embark on this adventure together!
Step 1: Distributive Property – Unleashing the Parentheses
The first hurdle in our equation-solving quest is the presence of parentheses. These seemingly innocent symbols hold within them expressions that need to be simplified before we can proceed further. This is where the distributive property comes into play, a powerful tool in the algebraist's arsenal. The distributive property allows us to multiply a term outside the parentheses by each term inside, effectively 'distributing' the multiplication across the expression. Let's see how this works in our case.
On the left side of the equation, we encounter 2(x-3). Applying the distributive property, we multiply the 2 by both the x and the -3, resulting in 2x - 23, which simplifies to 2x - 6. Similarly, on the right side, we have 3(x+1). Distributing the 3, we get 3x + 31, which simplifies to 3x + 3. Now, our equation looks a little friendlier: 2x - 6 + 9 = 3x + 3 + x. We've successfully navigated the first obstacle, and the path ahead is becoming clearer.
Step 2: Combining Like Terms – Gathering the Troops
With the parentheses out of the way, it's time to tidy up our equation by combining like terms. Like terms are those that share the same variable and exponent. In our equation, this means grouping together the terms with x and the constant terms (numbers without variables). This step is akin to organizing your workspace before tackling a big project, making the task at hand much more manageable.
On the left side of the equation, we have a constant term of -6 and another of +9. Combining these, -6 + 9 gives us +3. So, the left side simplifies to 2x + 3. On the right side, we have two terms with x: 3x and x. Remember that x is the same as 1x, so combining these gives us 3x + 1x = 4x. We also have a constant term of +3. Thus, the right side simplifies to 4x + 3. Our equation now stands as 2x + 3 = 4x + 3, a much cleaner and more approachable form.
Step 3: Isolating the Variable – The Quest for x
The heart of solving any equation lies in isolating the variable, in our case, x. This means manipulating the equation to get x all by itself on one side, revealing its value. To achieve this, we'll employ the golden rule of algebra: whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains balanced and the solution remains valid.
Our goal is to gather all the x terms on one side of the equation. Looking at 2x + 3 = 4x + 3, we see that the right side has more x terms (4x) than the left (2x). So, let's move the 2x from the left side to the right. To do this, we subtract 2x from both sides of the equation. This gives us 2x + 3 - 2x = 4x + 3 - 2x. On the left side, 2x - 2x cancels out, leaving us with just 3. On the right side, 4x - 2x simplifies to 2x, so we have 2x + 3. Our equation is now 3 = 2x + 3.
Next, we want to isolate the 2x term. To do this, we need to get rid of the +3 on the right side. We subtract 3 from both sides of the equation: 3 - 3 = 2x + 3 - 3. This simplifies to 0 = 2x. We're almost there!
Step 4: Solving for x – The Grand Finale
We've arrived at the final step in our equation-solving journey: solving for x. Our equation currently reads 0 = 2x. This tells us that 2 times x equals 0. The only number that, when multiplied by 2, results in 0 is 0 itself. Therefore, x must be 0. But let's formalize this with a final algebraic step.
To completely isolate x, we need to get rid of the 2 that's multiplying it. We do this by dividing both sides of the equation by 2: 0 / 2 = 2x / 2. On the left side, 0 / 2 is simply 0. On the right side, 2x / 2 simplifies to x. So, our final answer is x = 0. We've successfully navigated the equation and found the elusive value of x!
The Solution Unveiled: x = 0
After our meticulous journey through the equation 2(x-3)+9=3(x+1)+x, we've arrived at the solution: x = 0. This value, seemingly simple, is the key that unlocks the equation's hidden truth. But our exploration doesn't end here. To truly master the art of equation solving, it's crucial to verify our solution and understand its significance within the context of the original problem.
The Importance of Verification
In the realm of mathematics, accuracy reigns supreme. To ensure that our solution x = 0 is indeed correct, we must subject it to the ultimate test: verification. This involves substituting the value back into the original equation and checking if both sides balance. It's like double-checking your work before submitting a crucial report, ensuring that no errors have slipped through the cracks.
Let's plug x = 0 back into our original equation: 2(x-3)+9=3(x+1)+x. Replacing every instance of x with 0, we get 2(0-3)+9=3(0+1)+0. Now, we simplify each side independently.
On the left side, we have 2(0-3)+9. First, we tackle the parentheses: 0 - 3 = -3. Then, we multiply: 2 * -3 = -6. Finally, we add: -6 + 9 = 3. So, the left side simplifies to 3.
On the right side, we have 3(0+1)+0. Again, we start with the parentheses: 0 + 1 = 1. Next, we multiply: 3 * 1 = 3. Finally, we add: 3 + 0 = 3. The right side also simplifies to 3.
Behold! Both sides of the equation equal 3 when x = 0. This resounding confirmation assures us that our solution is correct. Verification is not just a formality; it's a cornerstone of mathematical rigor, a testament to the accuracy and reliability of our findings.
The Significance of the Solution
Now that we've verified our solution, let's ponder its significance. What does x = 0 really mean in the grand scheme of things? In the context of the equation 2(x-3)+9=3(x+1)+x, it signifies the point of equilibrium, the value that makes the equation a true statement. It's the single value that satisfies the relationship defined by the equation, a unique and precise answer.
Furthermore, understanding the solution provides insights into the nature of the equation itself. In this case, the equation is a linear equation in one variable. Linear equations, characterized by their straight-line graphs, have at most one solution. Our finding of x = 0 aligns perfectly with this principle, reinforcing our understanding of the equation's properties.
Real-World Applications: Where Equations Come to Life
Equations, like the one we've tackled today, are not just abstract mathematical constructs confined to textbooks and classrooms. They are powerful tools that underpin countless real-world applications, from the mundane to the extraordinary. Understanding how to solve equations is akin to possessing a universal key, unlocking solutions to problems across diverse fields.
Engineering and Physics: The Building Blocks of Progress
In the realms of engineering and physics, equations reign supreme. They are the language in which the laws of nature are written, the blueprints for technological marvels. From designing bridges that can withstand immense forces to calculating the trajectory of a spacecraft hurtling through the cosmos, equations are the bedrock of these disciplines.
Engineers use equations to model the behavior of structures, ensuring their stability and safety. Physicists employ equations to describe the motion of objects, the flow of fluids, and the interactions of particles. The equation we solved today, while seemingly simple, embodies the fundamental principles that underpin these complex calculations. The ability to manipulate and solve equations is an indispensable skill for anyone aspiring to a career in these fields.
Economics and Finance: Navigating the World of Money
The world of economics and finance is awash in numbers, and equations are the tools used to make sense of them. From predicting market trends to managing financial risk, equations play a crucial role in decision-making. Concepts like interest rates, inflation, and investment returns are all expressed through equations, allowing economists and financial analysts to quantify and analyze complex phenomena.
Understanding equations allows individuals to make informed financial decisions, whether it's choosing the right mortgage, planning for retirement, or investing in the stock market. The ability to solve equations empowers us to take control of our financial lives, making informed choices that shape our future.
Computer Science and Technology: The Digital Age
In the digital age, computer science and technology are at the forefront of innovation. Equations are the lifeblood of these fields, forming the basis of algorithms, software, and artificial intelligence. From the simplest computer program to the most sophisticated machine learning model, equations are the silent engines driving the digital revolution.
Computer scientists use equations to design efficient algorithms, optimize software performance, and create realistic simulations. Equations are also essential for data analysis, allowing us to extract meaningful insights from vast datasets. As technology continues to evolve, the importance of equations will only grow, making it a critical skill for anyone seeking to thrive in the digital world.
Conclusion: The Power of Equations
Our journey through the equation 2(x-3)+9=3(x+1)+x has been more than just a mathematical exercise. It's been a testament to the power of equations, their ability to encapsulate relationships, solve problems, and shape our world. We've seen how a seemingly complex equation can be broken down into manageable steps, revealing a clear and concise solution. We've also explored the real-world applications of equations, demonstrating their relevance across diverse fields.
So, the next time you encounter an equation, don't shy away from the challenge. Embrace the process of unraveling its mysteries, and you'll discover the power it holds to unlock new knowledge and possibilities. Keep practicing, keep exploring, and keep solving!