Solving The Linear Equation: 1/3(2x+1) - 2/5(x-2) = 3
Hey guys! Ever stumbled upon a seemingly complex equation and felt a bit lost? Don't worry, we've all been there! Today, we're going to break down and solve the equation 1/3(2x+1) - 2/5(x-2) = 3 step by step. This might look intimidating at first, but trust me, with a little bit of algebraic magic, we can conquer it. We'll focus on clarity and making sure you understand each stage, making the world of equation-solving a little less scary. So, grab your pencils, and let's dive in! This detailed walkthrough will not only provide the solution but also equip you with the skills to tackle similar problems with confidence. Understanding the basic principles of algebra is crucial for various fields, including science, engineering, and even everyday problem-solving. Let's embark on this mathematical journey together and unravel the mystery of this equation!
Breaking Down the Equation: Initial Steps
Alright, let's kick things off by taking a good look at our equation: 1/3(2x+1) - 2/5(x-2) = 3. The first thing we need to tackle is those fractions hanging out in front of the parentheses. Our mission? To get rid of them! To do this, we'll use the distributive property, which basically means we're going to multiply each term inside the parentheses by the fraction outside. Think of it as sharing the fraction love with everyone inside the parenthesis party! This step is crucial because it simplifies the equation and makes it easier to work with. By applying the distributive property, we expand the equation, setting the stage for combining like terms and isolating the variable x. This initial expansion is a fundamental technique in solving algebraic equations and helps to transform the equation into a more manageable form. Let's get started and watch the equation transform!
Applying the Distributive Property
So, let's distribute! First, we'll multiply 1/3 by both terms inside the first set of parentheses (2x and +1). Then, we'll multiply -2/5 by both terms inside the second set of parentheses (x and -2). Remember that negative signs are super important here, so pay close attention! This careful distribution is key to ensuring we maintain the integrity of the equation. Mistakes in this step can lead to an incorrect final answer, so precision is paramount. By correctly applying the distributive property, we eliminate the parentheses and create a linear equation that is easier to manipulate and solve. This step is like laying the foundation for the rest of the solution process. Let's see how it looks:
(1/3) * (2x) = 2x/3
(1/3) * (1) = 1/3
(-2/5) * (x) = -2x/5
(-2/5) * (-2) = 4/5
Now, let's put it all together. Our equation now looks like this:
2x/3 + 1/3 - 2x/5 + 4/5 = 3
See? We've already made progress! The fractions are still there, but at least they're not trapped inside parentheses anymore. We're one step closer to solving for x. This expanded form allows us to identify and combine like terms, which will further simplify the equation. The next step involves dealing with these fractions, which we'll tackle in the following section. Remember, each step we take brings us closer to unlocking the value of x!
Combining Like Terms: Simplifying the Equation
Okay, now that we've distributed and expanded our equation, it's time to gather our like terms together. Think of it like organizing your closet – you want to group similar items together to make things easier to find. In our equation, the like terms are the ones with x (2x/3 and -2x/5) and the constants (1/3 and 4/5). This step is all about making our equation more manageable. By combining like terms, we reduce the number of individual elements in the equation, making it simpler to solve for x. This is a crucial step in streamlining the solution process and avoiding unnecessary complexity. Let's get to it and see how much simpler we can make things!
Grouping x Terms and Constants
Let's start by grouping the x terms together and the constant terms together. This will help us visualize what we need to combine. It's like sorting your laundry before you wash it – you want to keep the socks with the socks and the shirts with the shirts! This organizational step is key to avoiding confusion and ensuring accuracy in the next step, where we'll actually combine these terms. By carefully grouping like terms, we set ourselves up for a smoother and more efficient simplification process. This is a fundamental technique in algebra and helps to clarify the structure of the equation. Here's how it looks:
(2x/3 - 2x/5) + (1/3 + 4/5) = 3
Finding a Common Denominator
Now, to actually combine these fractions, we need a common denominator. Remember those from math class? The common denominator is a number that both denominators (the bottom numbers in the fractions) can divide into evenly. For 3 and 5, the least common denominator is 15. Finding the common denominator is a critical step in adding or subtracting fractions. It ensures that we are working with comparable units, just like you can't add apples and oranges without first converting them to a common unit like