Solving The Equation (2/3)x - 2 = 7 A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the linear equation $\frac{2}{3}x - 2 = 7$. Linear equations are fundamental in mathematics and have widespread applications in various fields, including science, engineering, and economics. Understanding how to solve them is a crucial skill for anyone pursuing studies or careers in these areas. This article aims to provide a clear, step-by-step explanation of the solution, making it accessible to students and anyone interested in brushing up on their algebra skills. We will not only solve this specific equation but also discuss the underlying principles and techniques that can be applied to a wide range of similar problems. So, let's embark on this mathematical journey and unravel the solution together!

Understanding Linear Equations

Before diving into the solution, it's essential to grasp the concept of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is typically represented by letters like x, y, or z, and the highest power of the variable is 1. Linear equations can be written in the general form ax + b = c, where a, b, and c are constants, and x is the variable. The goal of solving a linear equation is to isolate the variable on one side of the equation, which means finding the value of the variable that makes the equation true. This involves performing algebraic operations on both sides of the equation to maintain equality while gradually simplifying it. Linear equations are the building blocks for more complex mathematical concepts, and mastering them is crucial for success in algebra and beyond. Understanding the properties of equality, such as the addition, subtraction, multiplication, and division properties, is key to effectively manipulating and solving these equations. In the following sections, we will apply these principles to solve our specific equation, $\frac{2}{3}x - 2 = 7$.

Step 1: Isolating the Term with the Variable

The first step in solving the equation $\frac{2}{3}x - 2 = 7$ is to isolate the term containing the variable, which in this case is $\frac{2}{3}x$. To do this, we need to eliminate the constant term on the left side of the equation, which is -2. We can achieve this by using the addition property of equality, which states that adding the same number to both sides of an equation preserves the equality. So, we add 2 to both sides of the equation: $\frac{2}{3}x - 2 + 2 = 7 + 2$. This simplifies to $\frac{2}{3}x = 9$. Now, we have successfully isolated the term with the variable on the left side of the equation. This step is crucial because it brings us closer to isolating the variable x itself. By adding 2 to both sides, we have effectively canceled out the -2 on the left, leaving us with a simpler equation to work with. This demonstrates a fundamental principle in solving equations: performing inverse operations to undo the operations that are applied to the variable. In the next step, we will focus on isolating x completely by dealing with the coefficient $\frac{2}{3}$.

Step 2: Solving for the Variable

Now that we have the equation $\frac{2}{3}x = 9$, the next step is to isolate x completely. The variable x is currently being multiplied by the fraction $\frac{2}{3}$. To undo this multiplication, we need to perform the inverse operation, which is division. However, dividing by a fraction can be a bit tricky, so instead, we can multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. So, we multiply both sides of the equation by $\frac{3}{2}$: $\frac{3}{2} * \frac{2}{3}x = 9 * \frac{3}{2}$. On the left side, the fractions $\frac{3}{2}$ and $\frac{2}{3}$ cancel each other out, leaving us with just x. On the right side, we have $9 * \frac{3}{2}$, which is equal to $\frac{27}{2}$. Therefore, the solution to the equation is $x = \frac{27}{2}$. This step demonstrates the power of using reciprocals to solve equations involving fractions. By multiplying by the reciprocal, we effectively eliminate the fraction and isolate the variable in one step. This is a common technique used in algebra and is essential for solving a wide variety of equations. In the next section, we will verify our solution to ensure its accuracy.

Step 3: Verifying the Solution

To ensure that our solution $x = \frac{27}{2}$ is correct, we need to substitute this value back into the original equation, $\frac{2}{3}x - 2 = 7$, and see if it holds true. Substituting $x = \frac{27}{2}$ into the equation, we get: $\frac{2}{3} * \frac{27}{2} - 2 = 7$. Now, let's simplify the left side of the equation. First, we multiply the fractions: $\frac{2}{3} * \frac{27}{2} = \frac{2 * 27}{3 * 2} = \frac{54}{6} = 9$. So, the equation becomes: $9 - 2 = 7$. This simplifies to $7 = 7$, which is a true statement. Since substituting our solution into the original equation results in a true statement, we can confidently conclude that our solution $x = \frac{27}{2}$ is correct. Verifying the solution is a crucial step in the problem-solving process. It helps to catch any errors that may have occurred during the solution process and ensures that the answer is accurate. This practice reinforces the understanding of the equation and the solution, making it a valuable habit for any math student. In the conclusion, we will summarize the steps we took to solve the equation and highlight the key concepts involved.

Conclusion

In this article, we have successfully solved the linear equation $\frac{2}{3}x - 2 = 7$ by following a step-by-step approach. First, we isolated the term with the variable by adding 2 to both sides of the equation. Then, we solved for the variable x by multiplying both sides of the equation by the reciprocal of the fraction $\frac{2}{3}$, which is $\frac{3}{2}$. This gave us the solution $x = \frac{27}{2}$. Finally, we verified our solution by substituting it back into the original equation and confirming that it resulted in a true statement. This process highlights the importance of understanding the properties of equality and using inverse operations to isolate the variable. Solving linear equations is a fundamental skill in algebra, and the techniques we have discussed here can be applied to a wide range of similar problems. By mastering these techniques, you will be well-equipped to tackle more complex mathematical challenges. Remember to always verify your solutions to ensure accuracy and build confidence in your problem-solving abilities. This comprehensive guide should serve as a valuable resource for students and anyone looking to strengthen their understanding of linear equations.