Solving For X: A Step-by-Step Guide To The Equation

Hey Plastik Magazine readers! Let's dive into a common mathematical problem: solving for x. Equations might seem daunting at first, but with a systematic approach, they become much more manageable. Today, we're going to break down the equation $-2(x+\frac{1}{3})+9x=4$ step-by-step. So, grab your calculators (or just your thinking caps!) and let's get started!

Understanding the Equation

Before we jump into solving, let's take a closer look at the equation. The equation we're tackling is $-2(x+\frac{1}{3})+9x=4$. Understanding the different parts of an equation is the foundation for success. We have variables, coefficients, constants, and operations. The variable in this case is x, which represents the unknown value we're trying to find. Coefficients are the numbers multiplied by the variable (like the -2 and 9 in our equation). Constants are standalone numbers (like the $\frac{1}{3}$ and 4). Finally, operations are the mathematical actions being performed (like addition, subtraction, multiplication, and division).

Now, let's talk about the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform calculations to ensure we arrive at the correct answer. Ignoring the order of operations is a common pitfall, so always keep it in mind! PEMDAS isn't just a suggestion; it's the law of the mathematical land. Understanding this order helps us tackle complex equations methodically, breaking them down into simpler, manageable steps. We will be applying this order throughout our solving process, ensuring accuracy and efficiency. So, let's keep PEMDAS as our guiding star in this mathematical journey.

Lastly, isolating the variable is the name of the game when solving for x. Our ultimate goal is to get x by itself on one side of the equation. This involves using inverse operations to undo the operations that are being performed on x. For example, if x is being multiplied by 2, we would divide both sides of the equation by 2. Similarly, if a number is being added to x, we would subtract that number from both sides. This is the core strategy, and we will be applying it strategically in each step.

Step 1: Distribute the -2

Our first step is to tackle the parentheses. Remember PEMDAS? Parentheses come first! We need to distribute the -2 across the terms inside the parentheses. This means multiplying -2 by both x and $\frac{1}{3}$.

So, $-2 * x = -2x$ and $-2 * \frac{1}{3} = -\frac{2}{3}$. This simplifies our equation to $-2x - \frac{2}{3} + 9x = 4$. Distribution is a fundamental technique in algebra, allowing us to simplify complex expressions. It's like unpacking a box to see what's inside. We're essentially unraveling the expression within the parentheses, making it easier to work with. A common mistake here is forgetting the negative sign. Always be careful with signs! They are like the secret agents of mathematics; a small oversight can change the whole mission. Double-check your signs at this stage to avoid errors down the line. When distributing, pay close attention to whether you're multiplying by a positive or negative number. A negative multiplied by a positive yields a negative, and a negative multiplied by a negative results in a positive. Mastering the art of distribution opens doors to solving a wide range of algebraic equations. It's a versatile tool in your mathematical arsenal, and understanding it thoroughly will significantly boost your problem-solving skills.

By distributing correctly, we've cleared the first hurdle in our equation-solving journey. The equation is now in a more manageable form, ready for the next step. We've successfully removed the parentheses and expanded the expression, paving the way for further simplification. With this foundational step completed, we can confidently move forward, building upon our progress to ultimately isolate x. So, let's keep up the momentum and tackle the next stage of the process.

Step 2: Combine Like Terms

Now that we've distributed, let's simplify the equation further by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, $-2x - \frac{2}{3} + 9x = 4$, we have two terms with x: -2x and 9x. We can combine these by adding their coefficients: -2 + 9 = 7. So, -2x + 9x becomes 7x. This simplifies our equation to $7x - \frac{2}{3} = 4$. Combining like terms is a crucial step in simplifying equations. It's like sorting your laundry – grouping similar items together makes the task much easier. This step not only simplifies the equation but also reduces the chances of making errors in subsequent steps. Imagine trying to solve an equation with a dozen terms scattered all over the place – it would be a recipe for confusion! Combining like terms brings order to the chaos, making the equation more visually and conceptually clear. This process reduces the complexity of the equation, making it easier to manipulate and ultimately solve for x. It's a fundamental skill in algebra, and mastering it will greatly enhance your ability to tackle more complex problems.

A common mistake is trying to combine terms that are not alike. You can only combine terms that have the same variable raised to the same power. For instance, you can't combine 7x with 4 because 4 is a constant term. It's like trying to add apples and oranges – they're just not the same! Remember, the variable part must match exactly for terms to be considered like terms. Always double-check the variable and its exponent before combining terms. This attention to detail will prevent errors and ensure the accuracy of your solution. Combining like terms is a powerful simplification technique. It streamlines the equation, making it more approachable and less intimidating. This step lays the groundwork for the next phase of our solution process, where we will further isolate x and get closer to our final answer.

Step 3: Isolate the Variable Term

Our next goal is to isolate the term with x on one side of the equation. Currently, we have $7x - \frac{2}{3} = 4$. To isolate 7x, we need to get rid of the $-\frac{2}{3}$. We can do this by adding $\frac{2}{3}$ to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. This is the golden rule of equation solving! Adding $\frac{2}{3}$ to both sides gives us $7x = 4 + \frac{2}{3}$. Isolating the variable term is like setting the stage for the grand finale. We're carefully positioning the pieces so that the final act – solving for x – can take center stage. This step is all about undoing the operations that are hindering x from being alone. We're peeling away the layers, one operation at a time, until only the x term remains on one side of the equation.

Maintaining balance in the equation is paramount. Think of an equation as a seesaw. If you add weight to one side, you must add the same weight to the other side to keep it balanced. This principle applies to all operations – addition, subtraction, multiplication, and division. Neglecting to maintain balance is a surefire way to derail your solution. Always double-check that you're performing the same operation on both sides of the equation. This will ensure that the equality remains intact and that your solution remains valid. We're systematically clearing the path for x to stand alone, and this meticulous approach is the key to our success. Each step brings us closer to unveiling the value of x, and isolating the variable term is a crucial milestone in this journey.

Now, we need to simplify the right side of the equation. To add 4 and $\frac{2}{3}$, we need a common denominator. We can rewrite 4 as $\frac{12}{3}$, so the equation becomes $7x = \frac{12}{3} + \frac{2}{3}$, which simplifies to $7x = \frac{14}{3}$.

Step 4: Solve for x

We're almost there! We now have $7x = \frac{14}{3}$. To solve for x, we need to get x completely by itself. Since x is being multiplied by 7, we need to do the inverse operation: divide both sides of the equation by 7. This gives us $x = \frac{\frac{14}{3}}{7}$. Dividing by 7 is the same as multiplying by $\frac{1}{7}$, so we have $x = \frac{14}{3} * \frac{1}{7}$. Solving for x is the grand finale of our equation-solving performance. It's the moment we've been working towards, the culmination of all our efforts. This is where we finally isolate x and reveal its true value. It's like the final piece of a puzzle sliding into place, completing the picture.

Let's talk about inverse operations! They are the key to unlocking the value of x. Every mathematical operation has an inverse operation that undoes it. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. By applying the inverse operation, we can systematically peel away the layers surrounding x until it stands alone. Think of inverse operations as the undo button in mathematics. They allow us to retrace our steps and isolate the variable. Mastering inverse operations is essential for solving equations. It empowers you to manipulate equations with confidence and precision.

Now, we can simplify the fraction. $\frac{14}{3} * \frac{1}{7} = \frac{14}{21}$. We can further simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 7. This gives us $x = \frac{2}{3}$.

Final Answer

So, we've successfully solved for x! The solution to the equation $-2(x+\frac{1}{3})+9x=4$ is $x = \frac{2}{3}$. The final answer is like the treasure at the end of a mathematical quest. It's the reward for our perseverance and problem-solving prowess. It's the moment of satisfaction when we can confidently declare,