Solving The Equation 5/6x - 3x + 4/3 = 5/3 A Step By Step Guide

Linear equations are the cornerstone of algebra, and mastering them is crucial for success in higher mathematics. In this comprehensive guide, we'll break down the process of solving the equation $\frac{5}{6}x - 3x + \frac{4}{3} = \frac{5}{3}$ step-by-step, ensuring you grasp every concept along the way. Let's dive in and conquer this equation together!

1. Understanding the Equation: Identifying the Key Components

To effectively solve the equation, it's essential to first understand its components. Our equation, $\frac{5}{6}x - 3x + \frac{4}{3} = \frac{5}{3}$, is a linear equation in one variable, 'x'. This means that the highest power of 'x' is 1. The equation consists of several terms: two terms containing 'x' ($\frac{5}{6}x$ and $-3x$), and two constant terms ($\frac{4}{3}$ and $\frac{5}{3}$). Our goal is to isolate 'x' on one side of the equation to find its value. Think of it like a puzzle – we need to rearrange the pieces to reveal the solution. Identifying the key components is the first step in this process. Recognizing the variable, the coefficients, and the constants sets the stage for the subsequent steps. Remember, linear equations represent a fundamental concept in mathematics, and mastering their solution techniques opens doors to more advanced topics. So, let's get comfortable with these components and prepare to tackle the equation head-on!

2. Combining Like Terms: Simplifying the Equation

Now that we understand the equation's components, let's combine like terms to simplify it. In our equation, $\frac{5}{6}x - 3x + \frac{4}{3} = \frac{5}{3}$, the terms $\frac{5}{6}x$ and $-3x$ are like terms because they both contain the variable 'x'. To combine them, we need to find a common denominator. The common denominator for 6 and 1 (since $-3x$ can be written as $\frac{-3x}{1}$) is 6. So, we rewrite $-3x$ as $\frac{-18x}{6}$. Now we can combine the 'x' terms: $\frac{5}{6}x - \frac{18}{6}x = \frac{-13}{6}x$. Our equation now looks like this: $\frac{-13}{6}x + \frac{4}{3} = \frac{5}{3}$. Combining like terms is a crucial step in simplifying any algebraic equation. It reduces the number of terms, making the equation easier to manipulate and solve. By grouping similar terms together, we create a more streamlined expression that highlights the relationship between the variable and the constants. This step not only simplifies the equation visually but also sets the stage for the next steps in the solution process. So, let's embrace the power of combining like terms and move closer to finding the value of 'x'!

3. Isolating the Variable Term: Moving Constants to One Side

Our next step in solving the equation is to isolate the variable term. This means we need to get the term containing 'x' (in our case, $\frac{-13}{6}x$) by itself on one side of the equation. To do this, we'll move the constant term $\frac{4}{3}$ to the right side of the equation. We can achieve this by subtracting $\frac{4}{3}$ from both sides of the equation. This maintains the equality because we're performing the same operation on both sides. So, we have: $\frac{-13}{6}x + \frac{4}{3} - \frac{4}{3} = \frac{5}{3} - \frac{4}{3}$. This simplifies to $\frac{-13}{6}x = \frac{1}{3}$. Isolating the variable term is a key strategy in solving equations. By strategically moving constants to the opposite side, we create a clearer path to isolating the variable. This step often involves using inverse operations, such as subtraction in this case, to effectively cancel out terms. The goal is to create a situation where the variable term stands alone, making it easier to determine the value of the variable itself. So, let's continue our journey towards solving for 'x' by isolating the variable term and setting the stage for the final step!

4. Solving for x: Multiplying by the Reciprocal

We're in the home stretch! Now that we have $\frac{-13}{6}x = \frac{1}{3}$, we need to solve for x. To do this, we'll multiply both sides of the equation by the reciprocal of $\frac{-13}{6}$, which is $\frac{-6}{13}$. Remember, multiplying a fraction by its reciprocal results in 1, effectively isolating 'x'. So, we multiply both sides: $\frac{-6}{13} * \frac{-13}{6}x = \frac{-6}{13} * \frac{1}{3}$. This simplifies to $x = \frac{-6}{39}$. We can further simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3. This gives us $x = \frac{-2}{13}$. Therefore, the solution to the equation is $x = \frac{-2}{13}$. Solving for x often involves using the inverse operation of multiplication, which is division. However, multiplying by the reciprocal is a more efficient way to achieve the same result, especially when dealing with fractions. This step demonstrates the power of using inverse operations to isolate the variable and reveal its value. So, let's celebrate our success in solving for 'x' and appreciate the step-by-step process that led us to the solution!

5. The Solution Set: Presenting the Answer

Finally, let's present our answer in the requested format: the solution set. The solution set is simply the set containing the value(s) of 'x' that satisfy the equation. In our case, we found that $x = \frac{-2}{13}$. Therefore, the solution set is {$\frac{-2}{13}$}. We can write this in the box as $\boxed{-\frac{2}{13}}$. The solution set is a concise way to represent the solution(s) to an equation. It emphasizes that we have found the value(s) that make the equation true. Presenting the answer clearly and accurately is just as important as the steps we took to solve the equation. So, let's take pride in our work and confidently present the solution set, knowing that we have successfully navigated the equation from start to finish!

In conclusion, solving the equation $\frac{5}{6}x - 3x + \frac{4}{3} = \frac{5}{3}$ involved several key steps: understanding the equation, combining like terms, isolating the variable term, solving for x, and presenting the solution set. By mastering these steps, you'll be well-equipped to tackle a wide range of linear equations. Keep practicing, and you'll become a pro at solving equations!